Volunteers and peshmerga forces carry their weapons north of Kirkuk, Iraq. (Stringer/Reuters)

FOR A year and more, Iraqi leaders and the U.S.-led coalition that joined them to fight the Islamic State ignored warnings that unless cleavages among Iraq's ethnic groups over territory and political authority were addressed, a victory over the terrorists would be followed by yet more civil conflict. Now, within days of the fall of the last major Islamic State-held Iraqi town, that fight may be beginning. A forceful move by Iraqi troops and allied Shiite militias into the city of Kirkuk on Sunday, accompanied by scattered fighting with retreating Kurdish forces, threatens to touch off a wider sectarian war.

The president of Iraq's Kurdistan region, Masoud Barzani, did much to provoke the intervention by the Iraqi federal government of Haider al-Abadi. Seeking to bolster his own political position, Mr. Barzani insisted on staging a referendum on independence last month despite Mr. Abadi's strong opposition and that of the United States, Kurdistan's closest ally. He then extended the vote to areas in and around Kirkuk, even though they are outside the Kurdistan autonomous region.

Kurdish forces now have been forced to yield key positions in Kirkuk, including the airport, a military base and several oil fields. In a sign of the internal disarray that should have forestalled Mr. Barzani’s referendum, some reports said that Kurdish fighters under the control of a political party opposed to him may have agreed to hand over the sites without opposition. U.S. officials, for their part, described the takeover as “coordinated movements, not attacks,” even though a number of people were killed in combat in and around the city.

Much will now depend on whether Mr. Abadi’s forces, and the Shiite militias under Iranian command, press their advantage or pause to negotiate. Kirkuk and its oil fields have long been disputed territory; never-implemented provisions of Iraq’s current constitution call for a referendum on its future status. That is one of the key disputes that Iraqi leaders dodged during the war against the Islamic State, even though it was clear that it would surface immediately afterward.

Other unaddressed troubles are festering. The Shiite-led government has done little to rebuild majority-Sunni cities destroyed in the war, including Fallujah and Mosul, and nothing to delegate more power to their leaders. The result is that Iraqi Sunnis, many of whom supported or tolerated the Islamic State for sectarian reasons, may be open to backing other Sunni militants, including al-Qaeda.

Mr. Abadi has long promised to address the country’s sectarian divisions and appears genuinely willing to do so, but he has been hamstrung by opposition from Shiite hard-liners with Iranian backing. Kurdish moderates who believed the region should repair its crumbling political system, or negotiate with Baghdad, were steamrolled by Mr. Barzani. The United States, which once worked hard to broker sectarian deals in Iraq, walked away from such hard work several years ago. Both the Obama and Trump administrations have claimed that the only U.S. interest in Iraq is destroying the Islamic State. The consequences of that blinkered view can now be clearly seen in Kirkuk.