From time to time, American troops fighting the Islamic State group in Iraq find themselves working alongside not only U.S.-trained Iraqi military personnel but also militias aligned with and reportedly armed by Iran.

These militias, most of them Shia Iraqis, have been part of the ISIS problem, not the solution. The atrocities they have committed against Sunni Iraqis – torture, summary executions, forced displacement, on top of mass arbitrary arrests by the government – helped push some Sunnis to welcome ISIS in the first place as an alternative to Baghdad’s heavy-handed and highly sectarian rule.

Click to expand Image Iraqi security forces and Shi'ite fighters sit in military vehicles near Falluja, Iraq on May 31, 2016. © 2016 Reuters

From a human-rights perspective, the issue now is not whether these militias are aligned with Iran; rather, it's their long and continuing record of abusive behavior toward Sunni Iraqis – and a tendency by some U.S. officials to downplay those abuses, especially in public.

A colleague and I recently met in Baghdad with several leaders of the Popular Mobilization Forces, or PMF, the body that unites the most significant militias under the nominal command of Iraq's prime minister. Yusif al-Kilabi, a PMF military spokesman, made the expected sorts of claims: that allegations of militia atrocities were exaggerated or pro-ISIS propaganda, and that in any event the crimes were the work of “a few individuals.”

He also made the fair point that atrocities against Shia Iraqis by ISIS and its precursor, al Qaeda in Iraq, played a key role in stoking the fierce sectarianism that is tearing Iraq apart.

Abu Mahdi al-Muhandis, the PMF's deputy commander, assured us that the most important militias now operate as one. But we came away unconvinced that such official status amounts to effective control. After we met with al-Muhandis, Kazhim al-Issawi, a leader of the Saraya al-Salam (Peace Brigades, what used to be the Mahdi Army), made clear that his organization answered to no one but themselves.

Al-Kilabi and al-Issawi separately claimed that their training now includes the basics of international humanitarian law. This may well be the case. But on the crucial question of accountability for serious crimes, essential to ending impunity for the worst atrocities, no PMF or other Iraqi officials could point to any instance of a fighter or commander disciplined or punished judicially for serious battlefield crimes. When we asked Prime Minister Haider al-Abadi why not, he shot back: “Do you want to see a civil war in the streets of Baghdad?”

The operation to retake Fallujah may have brought the issue to a head. On June 27, a bystander overheard a heated argument between army and militia commanders about whether the militias should remain in the city. The bystander told us he also heard counterterrorism commanders cursing the PMF, saying counter-terrorism forces were firing at PMF forces to try to keep them out of Fallujah.

Who are these militias?

After the fall of Mosul and disintegration of the Iraqi army in June 2014, and amid fears that ISIS might move in force against Baghdad, volunteer fighters mobilized under the PMF. This past February, Prime Minister al-Abadi declared the PMF to be “part of the Iraqi armed forces.”

The PMF's most important components are militias that have been around for years, even decades, nurtured by Iran. There are more than 50. Several of the largest – including armed wings of the Badr Organization, Peace Brigades, Hizbullah Battalions, League of Righteous – are responsible for attacking U.S. troops during the 2003-2011 Iraq war, in addition to the widespread killing of Iraqi Sunnis.

Human Rights Watch has documented numerous atrocities by some of these militias, including mass killings and torture, going back well before the rise of ISIS. We also documented extensive property destruction and forced relocation of Sunnis in areas that Kurdish Peshmerga fighters, closely allied with the U.S., recaptured from ISIS.

After the town of Amerli was retaken in August 2014, the Iraqi government restricted the militias' role in the March 2015 battle for Tikrit. The Iraqis were encouraged to do so by the U.S. But when regular military forces withdrew from Tikrit, PMF militias took over — free to loot and destroy homes.

In the current campaign for Fallujah, the PMF initially remained on the outskirts, but they’ve routinely detained escaping men and older boys for “screening.” Their screening methods have produced credible allegations of routine torture and extrajudicial killings.

It seems, too, that some militias are not content to remain on the city's outskirts. Fighters from the Badr Organization, for instance, have gone into the city as part of their day jobs, wearing federal police uniforms. Col. Christopher Garver, a spokesman for the U.S-led coalition fighting ISIS in Iraq, said that such “rumors” don't concern him “because we don’t support those groups. We support the federal police,” which is part of Iraq's Interior Ministry.

But for more than a decade, the federal police force has been a bailiwick of the same Badr Organization that allegedly abducted and killed scores of Sunnis. Witnesses to militia abuses outside Fallujah told us that federal police were among those beating and torturing them. They're suspected of executing at least 17.

And who holds them accountable?

As Colonel Carver’s comment illustrates, U.S. officials assert there is an operational difference between the PMF, now an official body, and the militias that in fact dominate the PMF. Retired Marine Corps Gen. John Allen, who served as President Obama's special envoy for the counter-ISIS coalition, explained to CBS News last year that there are “the militias that you and I are used to hearing about, that have close alignments with Iran” and “those elements, or the Popular Mobilization Force as they are known, [that] have been subordinated to the Iraqi higher military ... command.”

One might argue, as did the PMF we met in Baghdad, that the PMF is subordinate – on paper at least – to the prime minister's office. But essentially they are the same militias “that you and I are used to hearing” about. The PMF leaders we met in Baghdad said they understood the need for discipline within their ranks, but the Fallujah experience shows that remains aspirational at best.

General Allen’s successor as special envoy, Brett McGurk, has referred dismissively to “reports of some isolated atrocities committed by some of the Popular Mobilization Forces.” There have been hundreds of brutal beatings, instances of men being dragged behind cars, summary executions. The reports we collected make it hard to credit McGurk’s sense of “isolated.”

While speaking the media on June 10, McGurk rightly spoke about the need for accountability, insisting that “we’re talking to the right people almost 24/7 to try to make sure it’s handled appropriately.” A week earlier, Prime Minister al-Abadi said his government had opened an investigation and ordered the arrest of people responsible for “transgressions,” but there has been no information about who and how many have been arrested, and on what charges.

McGurk, responding to questions at a Senate committee hearing on June 28, said “about four to five members of the Iraqi army have been detained,” which suggests that so far the talk about accountability rings pretty hollow.

U.S. military and political leaders must do better and insist that those in the PMF who've committed crimes are made to account for them. It's disingenuous to promote the Iraqis' claims about investigations without ensuring that they actually yield results.