WASHINGTON — A top commander of the U.S.-led coalition battling the Islamic State group in Iraq and Syria has publicly acknowledged that Iraq’s Popular Mobilization Forces — a loose alliance of Iranian-backed Shiite militias — are “incidentally” benefiting from the extensive support provided to Iraqi security forces under the control of Baghdad.

Maj. Gen. Rupert Jones, deputy commander of the Combined Joint Task Force for Operation Inherent Resolve, noted that throughout the battle for Mosul, Iraq, and up until last week, the Popular Mobilization Forces, or PMF, were not deployed as an integral part of those Iraqi security forces directly supported by coalition operations. But with the Aug. 20 launch of the offensive on Tal Afar, the PMF is operating in the same sector alongside three Iraqi divisions, Iraq’s counterterrorism service, police units and the country’s emergency response division to liberate Tal Afar and the remainder of Nineveh province from ISIS.

“They’re all under the command of Prime Minister [Haider al]-Abadi. They’ve made a positive start, but we expect it to be a tough fight. And as always, the coalition will be there to support,” Jones told Pentagon reporters.

Estimated at some 20,000 fighters, the PMF was formally enshrined in a law signed late last year as part of the Iraqi security forces under the command of Abadi. It includes brigades of Iranian-backed Hezbollah and embedded operatives of the Quds force, the elite unit of Tehran’s Islamic Revolutionary Guards Corps.

At a meeting with Abadi in Tehran last June, Iranian Supreme Leader Ayatollah Ali Khamenei hailed the PMF as an “important and blessed phenomenon.” Conversely, Israeli leaders have assailed the force as a manifestation of Iran’s strategy to establish a land corridor that extends from the Arabian Gulf through Iraq and Syria and then onto Lebanon’s coast along the Mediterranean Sea.

In his Pentagon briefing, Jones noted that throughout the battle for Mosul, the PMF operated far west of the city under orders from Abadi. Today, he said, PMF fighters remain on the outskirts of Mosul, although he conceded that “elements may have slipped into the city … but you won’t find areas of the city held by the PMF.”

As for the battle for Tal Afar, which had been held by the PMF prior to the coalition-supported offensive, Jones insisted that U.S.-led support is indirect, yet inevitable due to what he called the congested battle space. “The Iraqi government made the decision that the PMF is to be integrated with elements of the Iraqi security forces. … So any support that we might provide to the PMF will be incidental to the fact that we are directly supporting the Iraqi Army, the counterterrorism force, the police and the like,” he said.

When asked about what specific types of support may indirectly benefit the PMF, Jones replied: “By ‘support’ I mean if you conduct an airstrike or a ground artillery fires in support of the federal police, and the PMF are operating on that same axis, they are going to get incremental support from it. Because without it, the Iraqi police can’t advance, nor can the PMF. … It’s always impossible to make a complete distinction in a congested battle space.”

The coalition commander noted that the Iraqi prime minister has used both the words “disciplined” and “nondisciplined” to describe the PMF, and warned against painting the entire PMF with a negative, Iranian-linked brush.

“There’s a risk in using a kind of blanket description of the PMF which creates a negative connotation. The PMF are now part of the Iraqi security forces. And forces which are moderate in their intent and forces that comply with the orders of Prime Minister Abadi rather than somebody else’s instructions are potentially a positive influence,” Jones said.

“We are here as a military coalition to support our partner forces in defeating [ISIS]. Not to do anything else,” he added. ”There are other forces and influences at play, but that is not part of the operation and not what our mission is about.”

Behnam Ben Taleblu, a senior Iran analyst for the Washington-based Foundation for Defense of Democracies, said incorporating the PMF into Iraq’s institutions of state security may ultimately threaten Iraqi sovereignty once ISIS is defeated militarily. “Iraqi sovereignty, I’m afraid, is like Swiss cheese, with a lot of the holes filled by Iranian influence.”

He insisted that U.S. policy and policy of other major coalition partners should be based on the political future of Iraq, after ISIS is defeated militarily. “We all should be wary of history repeating itself, where we win the battle militarily, yet lose it politically by allowing Iran to step in and fill the vacuum.”

When asked if the U.S.-backed coalition would be working directly with the PMF on so-called stabilization operations after the fighting is over, Jones deferred to political leaders in world capitals. “Do we have any freedoms at the moment to conduct that type of coordination? Absolutely not. And as a military man, I’d frankly be surprised if we received those sorts of instruction.”