Hashid Shaabi (Popular Mobilisation) allied with Iraqi forces against the Islamic State, carry their weapons prepare to attack Tikrit in the Iraqi town of Ouja, on the southern outskirts of Tikrit March 26. (Thaier Al-Sudani/Reuters)

When a video surfaced in January showing an American M1 Abrams tank flying the green-and-yellow flag of Kitaeb Hezbollah, the Iranian-backed militia in Iraq, U.S. officials were aghast.

The tank was filmed as ­pro-government fighters moved military equipment to Iraq’s western Anbar province ahead of an assault on the Islamic State.

American officials eventually determined that Kitaeb Hezbollah, which the United States considers a terrorist group, had not commandeered the tank. Rather, they concluded, a tank operator from the 9th Iraqi Army Division had raised the flag in a show of solidarity with Shiite militiamen joining them in the fight.

The message to the Iraqi government was clear: Militia flags need to come off all U.S.-supplied weaponry and equipment, and fast.

The incident illustrates the challenges that the Obama administration faces in Iraq, where government troops have increasingly fused into a single force with volunteer Shiite paramilitary fighters and Iranian-backed militias. The hybrid force makes it difficult for the United States to keep tabs on sensitive American-made military equipment.

In recent months, social media sites has carried a flurry of images depicting militia fighters carrying U.S.-made semiautomatic rifles or driving U.S. military vehicles draped with militia flags or insignia. Those photos are one byproduct of the renewed rise of militia groups funded and trained by Iran since the Islamic State swept across Iraq this past summer.

“U.S. weapons are on the battlefield — no question about it,” said Ahmed Ali, a senior fellow at the Education for Peace in Iraq Center. “They are in the hands of different groups, whether in the hands of the Islamic State or PMUs [popular mobilization units] or the Iraqi military.”

Since they were mobilized in the summer and put under formal government control, the largely Shiite volunteer forces known as popular mobilization units have fought alongside Iraqi forces and militias such as the Badr Organization and Asaib Ahl al-Haq, which launched thousands of attacks on U.S. forces during the last American war there.

Members of those same militias have been accused of sectarian abuses in areas where they have battled the Islamic State.

Ali said another militia, Kataeb al-Imam Ali, had been seen operating black Humvees typically used by Iraq’s elite counterterrorism forces. “The Iraqi government has no consideration for public relations” in dealing with the hard-line militants, he said. “It’s about the reality, and the reality is fighting ISIS,” which is another term for the Islamic State.

Such videos and images have created anxiety in Washington, where President Obama’s strategy to fight the Islamic State hinges on the abilities of Iraqis on the ground. Shown last year to be a decaying and shorthanded military, Iraq’s official security forces are being restored by U.S. and allied trainers but so far have failed to match the fighting power of paramilitaries also battling the Islamic State.

A State Department official, who like others spoke on the condition of anonymity to discuss talks with another government, said the Obama administration had in recent months reminded the government of Shiite Prime Minister Haider al-Abadi that such equipment must be used in line with U.S. arms export law.

“That means the Iraqi security forces should not lend or provide U.S.-provided equipment to militia forces without U.S. permission,” the official said.

U.S. officials, who think they have reasonably good visibility into the whereabouts of U.S. weaponry in Iraq, say there have been no confirmed cases of Iraqi security forces transferring ­U.S.-provided equipment to the sole custody of Shiite militias since this past summer.

Iraqi Defense Minister Khaled al-Obeidi said that “all weapons that the Americans are supplying are under the control of the Iraqi army.” In an interview, he said that U.S. officials had raised the issue of American weapons leaking into the hands of other groups but said that “it doesn’t happen.”

Yet keeping tabs on American weapons is a tall order, given the scale of U.S. military support to Iraq in the past dozen years. After President George W. Bush’s 2003 invasion, the United States spent more than $20 billion to rebuild and equip Iraqi security forces. Oil-rich Iraq also has become an important arms customer for U.S. defense firms.

Under U.S. law, the United States must conduct what is known as end-use monitoring of arms exports to ensure that U.S.-made weapons don’t go astray. While the frequency of such monitoring varies, U.S. officials stationed at the U.S. Embassy in Baghdad conduct checks on weapons such as armored fighting vehicles and missile systems “as security conditions allow,” said Lt. Col. Joe Sowers, a Pentagon spokesman.

Especially sensitive items, such as Abrams tanks, Stinger missiles and night-vision goggles, require “enhanced” monitoring, including annual inventories of equipment by serial number. On Sunday, according to one American official in Baghdad, U.S. personnel completed an inventory of Stinger missiles and Avenger systems, a surface-to-air weapon, that have been deployed to different parts of Iraq. All of the weapons were accounted for.

American officials suggest that militias may have acquired U.S.-made weapons, such as the M16A2 rifle, on the open market. But they also acknowledge that it is difficult on Iraq’s crowded battlefield to prevent weaponry or vehicles from being used by various forces.

The popular mobilization units meanwhile continue to occupy a gray area. Because they are paid by the Iraqi government and fall under government command, the Obama administration appears to have accepted that they will, at least sometimes, be using U.S.-provided weaponry, even though they are not formally part of the Iraqi security forces.

To complicate matters, the Abadi government has described the militias and Shiite volunteers as popular forces, raising the possibility that it will not distinguish between the two groups when it arms them.

According to Obeidi, the popular mobilization units have “their own simple weapons.” But Hadi al-Amiri, the influential leader of the Badr Organization, told The Washington Post last year that those forces received their weapons from the Iraqi government.

Some U.S. lawmakers are voicing doubts about how well a relatively small team at the U.S. Embassy in Baghdad can ensure the proper custody of U.S.-provided weaponry, especially given the limitations that U.S. personnel face in moving throughout the country.

“It increasingly seems like end-use monitoring is more of a goal than a reality,” said Sen. Patrick J. Leahy (D-Vt.), who authored a law prohibiting the United States from providing security assistance to forces linked to major human rights crimes. “That means we should expect some U.S. material to be used against our allies, and it may one day be used against us. More scrutiny and vetting is needed, but we also may need to consider if we can meet acceptable standards and whether it is an acceptable risk to our national interests.”

Oubai Shahbandar, a former Pentagon analyst who was stationed in Iraq from 2007 to 2008, said some U.S. officials had accepted questionable Iraqi assurances “at face value.”

“Not enough policy concern is being placed on the real likelihood that U.S. foreign military funds to the Iraqi government are enabling Iranian-backed militia forces,” he said.

The Obama administration is hoping that all paramilitaries will be brought under government control eventually, once the Iraqi parliament approves a plan to establish a national guard force that would include popular mobilization forces, militiamen and possibly Kurdish peshmerga.

Loveday Morris in Baghdad contributed to this report.