With Turkey seemingly victorious in Syria after the overthrow of Bashar Al-Assad last December, the stage is set for a potentially major escalation of rivalry between Ankara and Tehran in the region. In Iraq, the implications could be significant. The fall of Assad—a key ally in the Tehran-led ‘Axis of Resistance’—has shaped the ongoing debate over the dismantling of Iraq’s Popular Mobilization Units (PMU), a largely Shiite umbrella force backed by Iran. Such an outcome, alongside the loss of Syria as “forward defense,” could leave Iran with little wiggle room to rebuild its regional reach.

In this equation, Iraq’s Kurdistan region may emerge as a critical territory where Iran and Turkey could come to trade...