Jassem Abbas spent the past several months sleeping on the floor of a mosque that has become a scabies-infested refugee camp. But he won’t return to his once-comfortable home in Iraq’s northern city of Mosul even if it is liberated from Islamic Stateone day. His reasons highlight just how permanent Iraq’s divisions have become—and how difficult, if not impossible, it would be to put the country back together again. After more than a decade of internecine slaughter that followed the 2003 U.S. invasion, more and more Iraqis like Mr. Abbas are concluding that living together is no longer possible—and that the country’s three-way partition, formal or informal, may be the only viable option left.