U.S.-backed Iraqi troops are fighting side by side with Shiite paramilitary forces backed by Iran for the first time in the campaign to take back Mosul, deepening U.S. fears that Tehran is claiming a greater role in the critical battle to oust Islamic State militants from Iraq’s second-largest city.

Over the past week, Iraqi government units have quietly advanced alongside units of the Popular Mobilization Forces, the large coalition of Iraqi paramilitaries comprising primarily militias supported by Iran’s Revolutionary Guard Corps. They are jointly advancing on the northern Iraqi city of Tal Afar, less than 50 miles west of Mosul, Pentagon officials confirmed Monday.

Those units are trying to cut off possible escape routes for Islamic State fighters between western Mosul and the Syrian border, as Iraqi and Kurdish forces continue to press into the city’s northern, eastern and southern borders.

But this mingling of Iraqi government troops and the Shiite militias could put U.S. commanders in the uncomfortable position of having to provide American air support for militias with clear links to the Iranian regime, something Washington fiercely opposes.

U.S. commanders have made clear that no American assets would be used to assist or reinforce advancing militias fighting under the banner of the Popular Mobilization Forces, a U.S. defense official told The Washington Times. That policy will remain in place despite the presence of Iraqi troops within the militia’s ranks, the official said.

It is unclear how U.S. forces would respond if the Iraqi troops embedded with the militias called for American airstrikes, since American assets are tasked with defending Iraqi and coalition forces.

“That is going to be a challenge,” the official said.

Potential responses by U.S. forces to attacks on embedded Iraqi troops with the militias would not be limited to American fighters and bombers, a second U.S. defense official said. Armed American drones and U.S. artillery batteries in and around Tal Afar also could be used to defend Iraqi forces fighting alongside the militias, the official said Monday.

“The Iraqis have [jet] fighters too,” the official noted.

The militias, which are not officially part of the Iraqi armed forces but fall under the command of Prime Minister Haider al-Abadi, have played an integral role in the fight against the Islamic State, also known as ISIS or ISIL. They have played a major role in the recapture of the major Iraqi cities of Ramadi and Fallujah from the terrorist group’s control.

U.S. commanders in Iraq have deferred to Baghdad’s decision to incorporate the Shiite-dominated militia groups into the Mosul fight and remain optimistic that those forces will adhere to Iraq’s chain of command and not to their Iranian advisers.

But the militia’s victories, particularly in Fallujah, have been tainted by reports of egregious human rights abuses and sectarian violence against Sunni civilians by the Shiite groups allied with Tehran after the city fell.

Separately, Islamic State fighters on Monday appeared to be trying to divert the pressure building on their Mosul stronghold by launching suicide attacks in Fallujah and the Shiite holy city of Karbala.

Two suicide car bombs ripped through Fallujah’s city center, killing two Iraqi police officers and injuring 17 others, including civilians, The Associated Press reported. It was the first attack in the city since Iraqi forces liberated it from Islamic State control in June.

The attacks in Fallujah and Karbala, along with Islamic State assaults on Kirkuk, Sinjar and Rutba last month, forced Iraqi commanders to move troops from the Mosul operation to quell those attacks, which placed more responsibility on the militia’s shoulders.

But Baghdad’s decision to include Iraqi military units into the militia’s offensive was a positive sign that the paramilitary groups were refraining from such sectarian attacks in the Mosul offensive, Pentagon officials said privately.

Pressure from the Popular Mobilization Committee, the government-sanctioned commission tasked with overseeing the militias, as well as Iranian military advisers on the ground to avoid any sectarian violence, has resonated with both Sunni and Shiite militia leaders.

“Soleimani has put the fear of God into them,” regarding the consequences of any attacks on Sunni civilians by Shiite militias, said the official, referring to Iranian Maj. Gen. Qassem Soleimani, commander of the Quds force, the special operations wing of Iran’s military spearheading Tehran’s mission in Iraq.

“I think [the partnership with the Popular Mobilization Forces] is a positive step,” the official said.

U.N. spokesman Farhan Haq revealed Monday that some 54,000 civilians have fled their homes in the weeks since the formal campaign to recapture Mosul from Islamic State forces began last month, including more than 6,000 in the past four days. About three-quarters of the displaced people are being sheltered in camps set up by the United Nations and its humanitarian partners and one-quarter are been housed in host communities, the AP reported.

Mr. Haq said the World Food Program has provided assistance to more than 100,000 people fleeing the conflict, including a distribution to 25,000 people on Sunday in Gogjali, the first neighborhood retaken inside Mosul.

• This article is based in part on wire service reports.

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