BEIRUT/BAGHDAD (REUTERS) - Iraqi Shi'ite Muslim militias fighting in Syria have started returning home to combat a lightning advance by Sunni insurgents in their own country, an exodus that could affect the balance of power in Syria.

The movement demonstrates how intertwined the conflicts in the two neighbouring countries have become, as Sunni and Shi'ite fighters move between them and shifts in battlefield momentum on one side of the border influence the other.

President Bashar al-Assad has relied heavily on Iranian-backed Shi'ite militias from Iraq and Lebanon to help turn the tide against overwhelmingly Sunni rebels.

One of those militias, Iraq's Asaib Ahl al-Haq, an Iranian-backed group which fought U.S. troops during the 2003-2011 occupation, confirmed on Wednesday that it was pulling some of its fighters out of Syria.

Spokesman Ahmed al-Kinani said the group was responding to a call from Iraq's most senior Shi'ite cleric to protect Iraq from Sunni militants from the Islamic State of Iraq and the Levant (ISIL), who seized Iraq's biggest northern city last week. "Asaib Ahl al-Haq has withdrawn from Syria and returned to Iraq as both are religious duties to defend our sacred sites. Now they are fighting ISIL together with our security forces."

Estimates of the number of Shi'ite fighters in Syria have run in the thousands. Residents in Jaramana, a Damascus suburb with a heavy presence of government troops backed by foreign militias fighting rebels in the nearby district of eastern Ghouta, say fewer Iraqi fighters have been visible lately. "We easily differentiate them right away from their accents. You can also hear them down the street when they drive their cars blaring Iraqi Shi'ite music," said one Jaramana resident. "It's too soon to know for sure, but it seems their numbers have dropped in the past few days."

While the scale and impact of the redeployment is not yet clear, Syrian rebels have reported a cooling of clashes in eastern suburbs of Damascus, which some attributed to a withdrawal of Iraqi Shi'ite fighters supporting Assad.

On Friday, rebels set off a bomb in a building used by government forces near several major security sites near Abbasin Square on the eastern edge of Damascus - an act that would normally trigger a fierce counter-offensive. Instead, the rebels advanced to take several more buildings the next day. "To be honest, even the rebels are surprised at the lack of government reaction to this advance. Either (the government) is preparing for a very powerful counterattack, or they're weak at this moment," said Abu Yazan al-Shami, an activist based in east Ghouta, which borders Abassin Square.

Firas Abi Ali, an analyst at IHS Country Risk, said there was no evidence yet Iraqi fighters were withdrawing from Syria en masse. Any redeployment would probably be orderly, with coordination between Iranian and Syrian authorities, limiting the impact on the fight, he added in a note. "The government will compensate for any redeployment of Iraqi Shi'ite fighters, using manpower drawn from Hezbollah and other sources," Abi Ali said. "However, the Iraqi fighters' departure would probably temporarily reduce the ability of the Syrian government to mount new offensives and place it on the strategic defensive."

Syria's conflict has raised the profile at home of Iraqi Shi'ite militias, which have been helping the Shi'ite-led government fight ISIL for months around the edges of Baghdad and western Anbar province. Baghdad has been decorated for months with posters of Iraqi militia fighters slain fighting in Syria.

Rebels in Syria's eastern Deir al-Zor province along the frontier with Iraq say they have seen more flights to Iraq flying overhead over the past few days. "We're seeing five or six commercial airliners crossing into Iraqi air space from Syria every night," said Abu Hamza al Deiri, an activist based in Deir al-Zor.

However, some activists doubt the departure of Iraqi fighters will have much impact on the conflict which has killed more than 160,000 people and regularly kills 200 more per day. "We expect the Iraqis to leave, but I don't think that will happen massively. Plus, Hezbollah is sending more fighters to make up any shortage in case Iraqis withdraw," Damascus-based activist Susan Ahmad said.