"Iran killed an American contractor, wounding many. We strongly responded, and always will. Now Iran is orchestrating an attack on the US Embassy in Iraq. They will be held fully responsible. In addition, we expect Iraq to use its forces to protect the Embassy, and so notified," the tweet said.

The developments represent a major downturn in Iraq-US relations that could further undermine US influence in the region and also weaken Washington's hand in its maximum pressure campaign against Iran. Iraq has long struggled to balance its ties with the US and Iran, both allies of the Iraqi government.

Iraqi security forces made no effort to stop the protesters as they marched to the heavily-fortified Green Zone after a funeral held for those killed in the US airstrikes.

Dozens of protesters pushed into the embassy compound after smashing the gate used by cars to enter the embassy. The protesters, many in militia uniform, stopped in a corridor after about five metres, and were only about 200 meters away from the main building. Half a dozen US soldiers were seen on the roof of the main building, their guns were pointed at the protesters.

Smoke from the tear gas rose in the area, and at least three of the protesters appeared to have difficulties breathing. It wasn't immediately known whether the embassy staff had remained inside the main building or were evacuated at some point. There was no immediate comment from the US Embassy.

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The protesters hanged a poster on the wall: "America is an aggressor," and some commanders of militia factions loyal to Iran joined the protesters. Among those was Hadi al-Amiri, the head of the state-sanctioned paramilitary Popular Mobilisation Units, the umbrella group for the Iran-backed militias.

Yassine al-Yasseri, Iraq's interior minister, also appeared outside the embassy at one point and walked around to inspect the scene. He said that the prime minister had warned the US strikes on the Shiite militiamen would have serious consequences.

"This is one of the implications," al-Yasseri said. "This is a problem and is embarrassing to the government."

He said more security would be deployed to separate the protesters from the embassy, an indication the Iraqi troops would not move in to break up the crowd by force.

Earlier, the mob shouted "Down, Down USA!" as the crowd tried to push inside the embassy grounds, hurling water and stones over its walls and spraying graffiti.

US Secretary of State Mike Pompeo said Sunday's strikes send the message that the US would not tolerate actions by Iran that jeopardised American lives.

The Iranian-backed Iraqi militia had vowed Monday to retaliate for the US military strikes. The attack and vows for revenge raised concerns of new attacks that could threaten American interests in the region.

The US attack also outraged both the militias and the Iraqi government, which said it would reconsider its relationship with the US-led coalition - the first time it has said it will do so since an agreement was struck to keep some US troops in the country. It called the attack a "flagrant violation" of its sovereignty.

The US military said "precision defensive strikes" were conducted against five sites of Kataeb Hezbollah, or Hezbollah Brigades in Iraq and Syria.

AP with Reuters