His visit was drowned out in the American news cycle by feverish coverage of hearings on Capitol Hill regarding Russian meddling in last year's election. Sensing the moment, Abadi gestured to the national conversation, joking while sitting alongside Trump that he had nothing to do with wiretapping the then-candidate's phones. (Earlier in the day, FBI director James B. Comey had confirmed that his agency had found no evidence to support Trump's allegations about wiretapping.)

But the White House's relationship with the Iraqi government is hardly a laughing matter. As U.S.-backed forces and Iraqi government troops steadily take back the crucial northern city of Mosul from the Islamic State, attention is shifting to what happens once the battle is won. There are vexing challenges ahead: The weak Shiite-led government in Baghdad has yet to prove it has the ability to govern provinces where Sunnis comprise the majority; a movement for a referendum on an independent Kurdish state in northern Iraq is gaining traction; a host of other regional powers, including Turkey and Iran, are also exerting influence on the ground in competing ways.

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“We are proving that Daesh can be eliminated,” Abadi asserted Monday, referring to the jihadists of the Islamic State. He was speaking at the U.S. Institute of Peace, an agency of the government that the Trump administration plans to eliminate in its new proposed budget. Abadi offered a message of hope for his nation, arguing that things were looking up and that the country's fledgling democracy was successfully moving forward.

Still, critics, including human rights organizations, are concerned about the effects of an entrenched and deepening sectarianism. Abadi's Shiite-dominated government remains distrusted in areas reclaimed from the Islamic State. Key Iranian-backed Shiite militias mobilized by Abadi's government to fight in the anti-Islamic State campaign have been accused of carrying out their own massacres of Sunnis deemed to have collaborated with the extremist group.

A report to be aired by PBS Frontline on Tuesday details the mass disappearances of Sunni boys and men in a village outside of Baghdad once occupied by the Islamic State. Locals claim that the men were abducted by the Shiite fighters who had liberated the town from the jihadists. On a wider scale, as my colleagues have reported, the ineptitude of local officials and endemic graft among the police and judiciary in certain parts of Iraq have created room for Islamic State cells to return to provincial cities where they were only recently ousted.

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“We have inherited many problems, some of which are intrinsic in our society,” Abadi admitted during his talk at the USIP.

It's uncertain how the Trump administration will reckon with any of these problems. Ever since the election campaign, Trump's messaging on Iraq has been confusing. On one hand, he consistently decried the U.S.-led invasion of Iraq in 2003 (although it emerged that he was for the war before he was against it) and signaled that he wanted a radical break from his party's history of engineering regime change and embarking on nation-building projects in the Middle East.

But at the same time, Trump has pushed for a more muscular approach to fighting the Islamic State and exhibited an alarming disregard for Iraqi sovereignty with his perplexing calls to take the nation's oil.

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And then he decided to include Iraq on the list of seven Muslim-majority countries in the travel ban to the United States. (A second executive order removed Iraq from the list and faces renewed legal challenges in the courts.)

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“President Trump has talked a lot about defeating the Islamic State but done virtually nothing to address Iraq itself, except to lump it in with other suspect states in a temporary travel ban, a move he ultimately reversed amid protests from his own commanders, who objected to treating an ally with the back of the hand,” wrote Politico's Susan Glasser.

“While the Obama administration deserves blame for sidestepping Iraq’s political challenges, Mr. Trump has quickly exacerbated the trouble,” noted a February editorial in The Washington Post.

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The White House's proposed cuts to the State Department and its general apathy toward multilateral diplomacy don't inspire observers with much confidence.

“Trump’s efforts to 'deconstruct' the non-Pentagon parts of the U.S. foreign policy apparatus — as well as cuts to support for international institutions — threatens to strip the United States of the tools required in Iraq and elsewhere, just when we need them most,” wrote Jeff Prescott and Daniel Benaim, two former Obama administration officials, in Foreign Policy. “Unless we plan another occupation of Iraq (or genuinely and absurdly seek to 'take the oil') it is a reality that as the fighting stops our military will step aside and the State Department, USAID, the IMF, and the U.N. will have to take over.”

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And the situation on the ground is incredibly complex. Beyond rising Kurdish nationalist aspirations and the difficulty of reintegrating Sunni parts of Iraq, Glasser pointed out that Iraq's leader also has to deal with Baghdad's turbulent politics.

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“Abadi, who faces reelection next year, has much to worry about from within his own Shiite political party — as well as from the former prime minister, Nouri al-Maliki, widely seen as still waiting in the wings for Abadi to stumble,” Glasser wrote.

Trump has criticized his predecessors for gifting Iraq to Iran. But the Trump administration's antagonism toward Iran could alienate Abadi and other prominent Iraqi Shiites in Baghdad.

“Using Iraq as a battleground as part of a broader strategy to counter Iran would also ignore the foundation of America’s presence there — as the invited guest of the Iraqi government,” Prescott and Benaim wrote. "As much as Iraq needs us, we also need Iraq, particularly as we pursue persistent threats against the homeland — including as a hub for the continued fight against the Islamic State in Syria."

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In Washington, Abadi sought to show that his government is capable of being a solid partner to the United States. But the underlying tensions were not far from the surface.

“We have to work with others. We have to build bridges,” Abadi remarked at the close of his USIP talk. He ended with another joke at Trump's expense: “Otherwise, what do you do? You build walls?”