IRBIL, Iraq (AP) — As he stepped into office in 2005 to become Iraq's first Kurdish president, Jalal Talabani told his followers, "I am casting off my Kurdish clothes and wearing Iraqi ones instead. You must accept that." It was a symbolic call for unity: A longtime leader of Kurdish fighters, Talabani became the head of state of what was supposed to be a new Iraq, freed two years earlier from the rule of Saddam Hussein.

Talabani's death on Tuesday was a reminder of how that experiment in unity has frayed nearly to the point of unravelling: Only a week earlier, Kurds voted overwhelmingly in a...