Judy Woodruff: Now the final installment in our series Iran Rising in Iraq that examines Tehran’s influence there, and what it means for U.S. policy in the region. Washington is worried about that sway and presence in Iraq, and is taking measures to counter it, raising U.S.-Iran tensions. But, tonight, we look at an extraordinary moment when the U.S. and Iran made common cause to fight a common enemy, and why many say that is unlikely to happen again. In partnership with the Pulitzer Center on Crisis Reporting, here again is special correspondent Reza Sayah.

Reza Sayah: October 2016, a coalition of military forces in Iraq launched an offensive to take back the city of Mosul from ISIS. And fighting on the same side were the United States and the Islamic Republic of Iran. Iraqi Army General Ghais Al-Hamdawi says it was a superbly coordinated mission.

Maj. Gen. Ghais Al-hamdawi (through Interpreter): It was the perfect example of bravery and cooperation among everybody, the PMF, tanks, army, air force, the American Air Force, special ops, and even citizens took part. This battle should be a lesson for all the armies in the world.

Reza Sayah: The mission was called We Are Coming. Among the forces helping the Iraqi army, 500 American troops on the ground and U.S. fighter jets providing air support, and 16,000 fighters from the Popular Mobilization Forces, PMF for short, a volunteer Iraqi militia largely armed and funded by Iran and advised by Iran’s Revolutionary Guard. For the next several months, the Iranian-backed militia helped overpower ISIS on the ground in towns and villages surrounding Mosul. Once ISIS was encircled and trapped, in came Iraqi forces, backed by U.S. artillery units and airpower, to finish the extremist group.

Mazin Al-eshaiker: What unites both Iran and the United States is their goal to end ISIS, which we, as Iraqis, obviously appreciated.

Reza Sayah: Iraqi politician Mazin Al-Eshaiker says Washington and Tehran never publicly acknowledged the strategy and never made direct contact to discuss it.

Mazin Al-eshaiker: I’m talking the U.S. and Iranian didn’t sit face to face, but the Iraqis sat face to face with the Iranians, and, in the same token, sat face to face with the Americans to come up with a joint plan for both sides.

Reza Sayah: The plan worked. In July, ISIS was defeated in its last major stronghold, thanks in part to a rare occasion where the United States and Iran tacitly cooperated to beat a common enemy. But Iraqi officials say, don’t expect U.S.-Iran cooperation again in Iraq any time soon.

Mazin Al-eshaiker: We are free to dream what we want, but it will not happen.

Reza Sayah: Under President Barack Obama, the U.S. policy with Iran was cautious engagement on some issues. With the election of President Donald Trump, the policy immediately changed to confrontation, escalating the nearly four-decade-long cold war between the countries. In October, President Donald Trump repeated accusations that Iran sponsors terrorism in the region, and slapped sanctions against Iran’s Revolutionary Guard.

President Donald Trump: The Iranian dictatorship’s aggression continues to this day. The regime remains the world’s leading state sponsor of terrorism.

Reza Sayah: Mr. Trump also refused to certify that Iran was complying with the 2015 nuclear deal, even though the remaining world powers and U.N. inspectors said Iran was complying. Ten days later, U.S. Secretary of State Rex Tillerson visited Riyadh to boost Iraq’s ties with Iran’s main regional rival in the region, Saudi Arabia. Tillerson also suggested the PMF was an Iranian fighting force and called for the militia to disband, a demand the Iraqi government rejected, insisting PMF fighters were Iraqi nationals.

Rex Tillerson: Certainly, Iranian militias that are in Iraq, now that the fight against Da’esh and ISIS is coming to a close, those militias need to go home.

Reza Sayah: And, last week, CIA Director Mike Pompeo revealed that he had sent a letter to Qasem Soleimani, a senior commander in Iran’s Revolutionary Guard, warning Iran over its behavior in Iraq.

Mike Pompeo: He refused to open the letter. It didn’t break my heart, to be honest with you. What we were communicating to him in that letter was that we will hold he and Iran accountable for any attacks on American interests in Iraq by forces that are under their control. And we wanted to make sure he and the leadership in Iran understood that in a way that was crystal clear.

Reza Sayah: Senior Iranian officials have hit back in the war of words. Iran’s Foreign Minister Mohammad Javad Zarif called U.S. policy in the Middle East dangerous. In a live televised address, Iranian President Hassan Rouhani accused Western countries, including the United States, of having fed and armed ISIS. And in a speech to university students last month, Iran’s supreme leader called the U.S. Iran’s number one enemy.

Ayatollah Ali Khamenei (through Interpreter): My dear children, don’t forget that in this very important path where you’re following your goals, your number one enemy is America.

Seyed Hosseini: America has to learn a lesson.

Reza Sayah: Iran-based pro-Iranian political analyst Seyed Hosseini says better relations between Washington and Tehran is good for Iraq. But that won’t happen, he says, unless the U.S. changes what Hosseini calls a hostile policy against Iran.

Seyed Hosseini: Until they don’t correct themselves and their policies in the region, I don’t think there will be a great hope for that. America, for them to be present in the region, they need Iranian help. They must just come to terms and accept the presence of a powerful Iran.

Reza Sayah: Many Iraqis doubt Tehran and Washington will change their policies. Ali Elami has owned this Baghdad supermarket for five decades. This is where Iraq’s former dictator Saddam Hussein used to stop by for late-night shawarmas, he says. So, Saddam Hussein had shawarma at your place? Elami says the U.S. and Iran are both here for their own interests, not to help Iraq.

Ali Elami (through Interpreter): The location of Iraq is very strategic. There’s oil, rivers. When Americans came and kicked out Saddam, they didn’t do it for our interests. They did it for oil and money. Iran has expanded here not for our sake. They did it for their own benefit.

Muthanna Amin Nader: We pay a price as a people in Iraq.

Reza Sayah: Iraqi politician Muthana Amin Nader is happy to see is defeated in Iraq. But what he fears now is a dangerous proxy war between Iran and the U.S.

Muthanna Amin Nader: Conflict between Iran and America makes our people as victim. We give a very, very expensive price. It’s time to say enough for bleeding in Iraq and destroying Iraq. They should support us, but also keep away from us.

Reza Sayah: With so much at stake here for the U.S. and Iran, keeping away from Iraq seems unlikely. How the two adversaries manage that high-stakes competition while they’re here may go a long way in shaping the future of Iraq. For the PBS NewsHour, I’m Reza Sayah in Baghdad.