A portrait of Turkish President Recep Tayyip Erdogan hangs on a building near Taksim Square as Erdogan supporters wave Turkish flags in Istanbul on Sunday. (Kemal Aslan/Reuters)

This post was updated at 7:59 a.m.

In the uproar following the attempted military coup in Turkey, relations between Washington and Ankara, already badly strained, appear to be headed for new difficulty.

The immediate test will be Turkish President Recep Tayyip Erdogan’s demand that the United States extradite Fethullah Gulen, an exiled Turkish Muslim leader blamed by Erdogan for the coup. Obama administration officials say they will consider any such request, under an existing extradition treaty with Turkey. But the issue will be delicate, given past U.S. and European criticism of Erdogan’s human rights record.

Turkish indignation over Gulen’s residence in the United States was voiced Monday by Egemen Bagis, a Turkish politician and longtime adviser for Erdogan, in a telephone interview from Turkey and a subsequent email.

“America can no longer drag its feet in the sand,” Bagis said. He called Gulen “a traitor to his country” who “through his network of henchmen was able to instruct and encourage a military coup.” Bagis warned that if the United States ignores the request for Gulen’s extradition, it will “run the risk of losing an ally.”

Suleyman Soylu, Turkey’s labor minister, even claimed on Saturday that “the U.S. is behind the coup attempt.” Such comments put Secretary of State John F. Kerry in the unlikely position of having to reject what he called “utterly false” reports of U.S. complicity with the plotters.

Gulen has denied any role and asserted Saturday that “it could be a staged coup” to allow a crackdown against his movement. There’s no evidence to support this claim. Anger at the coup plotters is shared by all the major Turkish political parties, including the Kurdish party known as the HDP. “The worst politician is better than the best general,” a prominent Kurdish activist told me Monday in condemning the coup.

The tensions in the U.S.-Turkey relationship go much deeper than the issue of Gulen’s alleged role in the coup and the demand for his extradition. The two countries are sharply divided over strategy for combating the Islamic State in Syria. Erdogan’s government argues that the United States has allied with a Syrian Kurdish group known as the YPG that has links with the underground Kurdistan Workers’ Party, which the Turks call a terrorist organization.

“We have a serious crisis in U.S.-Turkish relations,” argued Bulent Aliriza, a leading Turkish analyst at the Center for Strategic and International Studies in Washington, in a telephone interview Monday from Ankara. Aliriza notes that because of the bitter divide over Syria policy, the relationship “was troubled even before the coup.”

Because Turkey is a NATO ally, many in Washington had assumed that Ankara would cooperate in the U.S.-led strategy against the Islamic State. But Aliriza cautioned that the U.S. mistakenly assumed that Erdogan would allow Turkey to be a base for the U.S. invasion of Iraq in 2003. The U.S.-Turkey split over the YPG “is potentially more divisive than the Iraq decision,” Aliriza warned.

The post-coup deterioration in relations was signaled when Turkey halted U.S. bombing operations against the Islamic State from Incirlik Air Base in southern Turkey. Ankara’s agreement a year ago to allow a squadron of A-10 “Warthog” ground-attack planes to fly from Incirlik had demonstrated renewed cooperation between the two countries’ militaries after a decade of tense relations— and according to U.S. and Turkish analysts, had bolstered the Turkish military after a decade of increasing political control.

Any independent role for the military was shattered by Friday’s botched coup. Some Turkish analysts have pointedly noted that some elements of the Turkish Air Force, allegedly including jets operating from Incirlik, were part of the plot. A Turkish Air Force general who requested asylum from the U.S. garrison at Incirlik was handed back by the Americans there.

The recent history of U.S.-Turkey relations offers a lesson in how a friendship can unravel. Back in 2011, after the so-called “Arab Spring” toppled authoritarian rulers in Tunisia, Egypt and Libya, President Obama developed what amounted to a political partnership with Erdogan. The White House believed that his ruling Justice and Development Party was a model for how Islamist parties could coexist with democracy and free markets.

But the relationship soured over the past two years as Erdogan tightened political control of the Turkish press and judiciary and balked at U.S. strategy for fighting the Islamic State. Obama pointedly criticized Erdogan in April, after he visited the White House, saying that his recent crackdown on journalists “could lead Turkey down a path that would be very troubling.” A peeved Erdogan said that he was “saddened” by such criticism.

Because of Erdogan’s concern about Turkey’s isolation, he had moved this summer to rebuild his relations with Israel and Russia, two countries with which he had been feuding. Ankara’s relationship with Washington is even more important, to both countries, but it won’t be easy to repair.