US troops fired tear gas to disperse a crowd that breached the outer wall of the compound, which is in the capital's heavily fortified Green Zone.

One of the guard towers was also apparently set alight.

At least 25 fighters died when the US bombed bases associated with Kataib Hezbollah in Iraq and Syria on Sunday.

The US blamed the militia for a rocket attack on an Iraqi military base in Kirkuk on Friday that killed an American civilian contractor.

Iraqi Prime Minister Adel Abdul Mahdi said the air strikes had violated his country's sovereignty and would force it to review its relations with the US.

Kataib Hezbollah's commander, Abu Mahdi al-Muhandis, warned that the militia's response "would be very tough on the American forces in Iraq".

Iran described the attacks as a "clear example of terrorism".

Tuesday's protest took place after funerals were held in the capital for militia fighters who were killed in the US strikes.

Thousands of mourners - including several senior Iran-backed paramilitary and militia leaders - marched towards the Green Zone, where many Iraqi government offices and foreign embassies are located.

Several senior paramilitary and militia leaders were among the protesters

They were allowed by Iraqi security forces to enter the zone and gather on a street outside the US embassy compound.

Waving Kataib Hezbollah and other militia flags, and chanting anti-American slogans, the protesters threw stones at the compound's main gate, pulled down security cameras, attacked empty guard posts, and started several fires.

The situation then escalated when the compound's wall was breached.

The Associated Press reported that a gate at a side-entrance to the compound was smashed open and that hundreds of people pushed about 5m (16ft) into a corridor leading to the main embassy building before being forced to retreat by tear gas fired by US troops.

It was not clear whether there were any staff inside the embassy at the time. There was an unconfirmed report that the ambassador had been evacuated.