BAGHDAD — Shi’a militiamen, supported by a smaller group of government troops, advanced on Tuesday to within a few kilometers of a university on Ramadi’s southwestern edge, police sources and Sunni tribal fighters allied to the government said.

As they passed through farmland south of Ramadi, the militiamen told people to return home and stay inside, promising they would not be harmed.

The loss of Ramadi a week ago was swiftly followed by the fall of the UNESCO heritage city of Palmyra in Syria, the two biggest gains by “Islamic State” fighters since the United States began air strikes on them in Iraq and Syria last year.

ISIS controls swathes of territory in both countries, where it has proclaimed a caliphate to rule over all Muslims according to strict medieval precepts.

In Iraq, the government’s failure to hold Ramadi has forced Baghdad to send in the Iran-backed Shi’a paramilitaries. Washington fears this could enrage residents in the overwhelmingly Sunni province and push them into the arms of ISIS.

A spokesman for the Shi’a militias, which are known as the Popular Mobilization Forces (PMF), said Tuesday the codename for the new operation would be “Labaik ya Hussein”. This is a slogan in honor of a grandson of the Prophet Mohammed killed in a seventh century battle that was one of the causes of the schism between Shi’a and Sunni Muslims.

But the operation was rebranded “Labaik ya Iraq” (We answer your call Iraq) after Iraqi and Western criticism to the sectarian tone of the initial name.

In Washington, Pentagon spokesman Colonel Steve Warren said the key to victory would be a unified Iraq “that separates itself from sectarian divides, coalesces around this common threat”. Asked about the sectarian codename, he said: “I think it’s unhelpful.”

The militia fighters have performed better than Iraq’s own army, but their presence risks alienating Sunni residents, especially if they emphasize sectarian aims.

Just weeks ago, the Baghdad government and allied militia appeared to be having success against ISIS, recapturing former dictator Saddam Hussein’s home city of Tikrit.

But Anbar has proven more difficult. PMF had until now stayed out of the area, where Sunni tribes have been hostile to outsiders for generations.

The United Nations expressed concern that civilians trying to flee Ramadi were being halted at police checkpoints, forcing them to return to the combat zone.

Washington hopes the government can win the support of Sunni tribal fighters, a tactic U.S. Marines used in Anbar to defeat al Qaeda during the 2003-2011 occupation of Iraq.

The Baghdad government has succeeded in persuading some Sunni tribal leaders to accept help from the Shi’a fighters, but mistrust runs deep after years of sectarian war in which atrocities were committed on both sides.

The commander of the Quds Force of Iran’s Revolutionary Guards, the main Iranian force backing its allies abroad, mocked Washington for doing too little to help Baghdad.

“Obama has not done a damn thing so far to confront Daesh. Doesn’t that show that there is no will in America to confront it?” Qassem Soleimani said, using an Arabic acronym for Islamic State.

His remarks came a day after U.S. Defense Secretary Ash Carter infuriated Baghdad by saying the Iraqi army had abandoned Ramadi because it lacked “the will to fight”, remarks Iraqi Prime Minister Abadi said showed Carter was misinformed.

In a move of apparent damage control, Vice President Joe Biden phoned Abadi on Monday to reassure him Washington still supported Baghdad.

Carter said on Thursday Pentagon officials have begun to examine how the U.S. military could better equip and train Iraqi troops.

Carter told reporters on his plane to Asia that he had convened a group of defense policy officials and military officers from U.S. Central Command and the Pentagon’s Joint Staff to look at how “we can enhance, hasten” the mission to train and equip Iraqi forces.