It is October 2018 and European Council President Donald Tusk pronounces that there is no more to be said: The leaders of 27 European Union member states are ready to make their final offer. The call goes out to the British delegation, where Prime Minister Theresa May is huddled with her advisers, preparing for the endgame of Brexit.

Mrs May makes the short walk to the soundproof conference room and takes her seat. Her face betrays no emotion.

In the coming hours, Mrs May will determine her country's political and economic destiny. There will be no sleep for her or her counterparts, Mr Tusk and chief EU negotiator Michel Barnier, who have decreed that if there is to be a deal it must be done now.

European leaders are barely acquainted with the British Prime Minister, who does not specialise in personal relationships. They stare at her for clues of her intentions but there are none. Mrs May sits silently and listens as the EU's final Brexit offer is set out. As night turns to day in Brussels, British and European history will be made.

Mrs May knows if she pushes too hard, the EU could halt the talks, putting her at the cliff edge in 2019: An exit with no deal. Tariffs, border controls and massive economic disruption would follow. British businesses might never forgive her, Parliament could rise up against her - the clamour for Scottish independence and the break-up of Mrs May's "precious" United Kingdom might become irresistible.

But give too much ground and Mrs May knows all too well what will follow. Europe has played a key role in the downfall of the past three Conservative prime ministers, and her party will show no mercy if she sells Britain short now. As Sir John Major, the former prime minister, said of Mrs May's Eurosceptic critics last month: "They may be allies of the Prime Minister; the risk is that tomorrow they may not (be)."

It is commonplace to say that few people really know Mrs May, let alone her bottom lines for Brexit. But there are clues to be found by examining her record as a negotiator as Britain's home secretary and by speaking to those who have worked alongside her and sat on the opposite side of the negotiating table. Mrs May likes to be in control; she compromises only when all other options are exhausted and she deals in facts - not personal relationships. Any deal may not take shape until very late in the day.

Of course Mrs May might never have her rendezvous with political destiny in the European Council's "space egg" Europa headquarters in autumn next year: The negotiations could have collapsed long before then. Mr Barnier wants an early deal on Britain's exit bill and a fight over money could derail the talks. But working on the usual EU principle that nothing is agreed until everything is agreed, money, market access, border controls and a transitional arrangement could still be in play when Mrs May begins the ultimate negotiation.

Over the next 18 months, the talks will cover a bewildering range of subjects from aviation to nuclear safety but in the end Mrs May will have to make a small number of strategic trade-offs.

She has already announced she wants a "hard Brexit": out of the single market, the Customs union and the jurisdiction of the EU court. But she has also committed to "the freest and most frictionless trade possible in goods and services between the UK and the EU".

What is she prepared to offer to get it? A liberal immigration regime for EU citizens? A limited continuing role for the European Court of Justice? Billions of pounds for the EU budget?

CRAB-LIKE PROGRESS

Experience suggests that Mrs May will play it long. Mr Frans Timmermans, who oversees home affairs and migration for the European Commission, says the British leader has "an admirable crab-like way of going at things".

Mr Timmermans, who dealt with Mrs May during her six-year spell as home secretary, adds: "She moves calmly and sideways, she takes her time, but once she gives her word she sticks to it. That is one of the things I really appreciate about her. In these negotiations we will need this kind of integrity on all sides. For that is the only way we can ensure that we do the least harm possible in what - by its nature - unfortunately is going to be a painful process."

Whitehall officials confirm it is normal for Mrs May to take decisions in the early hours after careful study. One of the most remarkable decisions of her time at the Home Office came in 2012 when she decided not to extradite Mr Gary McKinnon, accused by the US of hacking government computers. Mr McKinnon has Asperger's syndrome, but the US was furious.

"There were piles and piles of papers around - they were reviewed and reviewed," says one close observer of Mrs May at the Home Office. "The Americans had convinced themselves he was going to be extradited. She said I'm going to go through all the papers again. She reached a decision at some time in the night." The plane was ready to take Mr McKinnon to the US and a possible 60-year jail sentence.

Mrs May relies more heavily on briefing packs than personal relationships when it comes to deal-making. Uncomfortable footage of her at a recent European summit awkwardly standing on her own while colleagues gossiped in the background seemed to sum up her "outsider" position in the club.

The Prime Minister's efforts at diplomacy can be clunky. At a recent meeting with German Chancellor Angela Merkel, Mrs May suggested an early resolution of the issue of citizens' rights after Brexit. Dr Merkel's team had already indicated it was too early to talk about the issue, which she saw as part of the wider negotiation. One official at the meeting says it was "frosty and difficult".

By contrast, former Tory prime minister David Cameron put a special focus on wooing Dr Merkel, hosting the German Chancellor at cosy weekends in his Chequers country retreat where they watched DVDs and went for walks. It was not entirely successful: Dr Merkel ultimately failed to help Mr Cameron on a range of issues, including EU reform.

Head of the Centre for European Reform Charles Grant says: "Arguably Cameron invested a bit too much in Merkel but there's a view in Whitehall that she has underinvested in Berlin."

While Mr Cameron could also call on Mr Mark Rutte and Mr Fredrik Reinfeldt, the Dutch and Swedish prime ministers, in a tight corner, Mr Grant says: "I don't think she (Mrs May) has those personal relationships at the moment."

Mrs May's view is that in the end Brexit will be determined by cold calculations of national interest, not diplomatic schmoozing. Sir John is not so sure: "The atmosphere is already sour," he says. "A little more charm and a lot less cheap rhetoric would do much to protect the UK's interests."

The Prime Minister is also regarded by counterparts as an unyielding negotiating partner: utterly straight but not flexible enough to find compromises on the hoof. In the intense atmosphere of the final round of Brexit talks, some believe this could be a serious flaw.

Mr Nick Clegg, Britain's former Liberal Democrat deputy prime minister in Mr Cameron's coalition government, clashed repeatedly with Mrs May in the Cabinet on issues including immigration and civil liberties and came to respect her for her seriousness.

But he says her skills as a negotiator at the Home Office and Number 10, where she has control of most of the variables, may not be transferable to the "Rubik's cube reality" of negotiating with 27 other member states.

"In our negotiations in the coalition she could be relied on to stick to what we had agreed, though it was invariably an unnecessarily laborious process to get to an agreement in the first place," he says.

"But she will now need to develop quite different quicksilver skills - ingenuity, agility, an ability to think on her feet - when dealing with 27 other governments and Parliaments, each with their own needs and bugbears."

One minister who worked with Mrs May at the Home Office notes she rarely makes decisions in front of colleagues: Rather, she listens to arguments and then goes away to make her own conclusion. "She listens to a presentation; she will let everyone else interject but she rarely makes a decision in a meeting," the minister says. It is a time-consuming approach which may rankle with sleep-deprived leaders in the final stages of Brexit talks. Her apparent intransigence could push matters to the brink.

A former police chief who used to negotiate regularly with Mrs May at the Home Office says: "Sitting in front of Theresa you can feel very inadequate. Having a disagreement with Theresa is chilling. There were times when I thought I could sit with Theresa May for 24 hours a day for the rest of time and I still wouldn't manage to convince her that her position is wrong."

Another former police leader says that after one dispute with Mrs May it had become clear she would not move. "In the end, I had a call from David Cameron and he agreed to a new position which I was happier with. She had been unable to retreat herself, and her boss had to do it for her. Who will perform this function now that she is the Prime Minister?"

WINNING BRUSSELS OVER

But Mrs May has shown some flexibility in Europe, winning admiration in Brussels and Westminster for the way in which she negotiated Britain's continued membership of several EU judicial and crime-fighting policies - including the European arrest warrant - when Tory Eurosceptics were clamouring for her to exercise a right to opt out of all of them.

Among her fellow European interior ministers - something of a policy world-apart in Brussels - she developed a strong reputation: well briefed, methodical and someone who would stick to a deal.

"I had excellent relations with Mrs May for a very specific reason: her seriousness," says France's Prime Minister Bernard Cazeneuve, a former interior minister. "She is someone serious, meaning there was competence and frankness in our relations. To have an able interlocutor is always preferable to having one who isn't."

Her meetings were short on small talk and long on detail. And she showed little fear for awkward diplomatic moments. When negotiations became sticky, Mrs May would readily deploy "a Pinteresque silence" to drive home her point, according to one witness.

But one senior EU figure says he feared Mrs May had drawn the wrong lesson from that negotiation, which allowed Britain to "cherry pick" the judicial policies in which it wanted to participate. Dr Merkel has already ruled out a "cherry picking" approach to Brexit.

Given Mrs May's preference for sticking to her positions and the fact she is not known for nimble political footwork, she will already have in mind where her final negotiating "flex" will come from when she walks into the European Council for the final time.

Clues can be gleaned in the government's Brexit White Paper which is specific in some areas but virtually silent in others, noticeably on Britain's future payments to the EU budget and its new EU work permit regime. Mrs May has ruled out "vast" payments but Tory Eurosceptic MPs privately indicate they would be prepared to pay something for market access.

A generous immigration regime would also help, though it would be completely against Mrs May's instincts to trade border controls for economic gain. As former foreign secretary William Hague puts it: "The closer you are to free movement, the closer you can be to the single market."

For now Mrs May is hanging tough, insisting that "no deal is better than a bad deal" and threatening to walk out of talks if the terms are not right. Foreign Secretary Boris Johnson says leaving the EU on World Trade Organisation terms would be "perfectly OK". The prospect alarms business, however. A breakdown in talks in October next year would trigger market convulsions.

Mr Clegg says: "I don't blame her for striking a defiant and confident tone ahead of the talks - what else can she do? But I think she made a huge error in setting wildly unrealistic expectations about what can be achieved in a very short timeframe."

Is she bluffing about walking away? "No," says former EU trade commissioner Peter Mandelson. "There's absolutely no evidence of that. I've spoken to senior people in government about this. It's not a pose."

One minister who worked with Mrs May at the Home Office says no one can guess what the Prime Minister is thinking: "She's the best poker player I've seen." When Mrs May takes her seat in the Europa building for the final round of Brexit talks, she will be playing for the highest stakes imaginable.

THE FINANCIAL TIMES

•Additional reporting by Helen Warrell and Jim Brunsden