In April, Iraqi Prime Minister Haider al-Abadi sat at a conference table in his Baghdad office with almost two dozen men in combat fatigues. The men were not officers in the Iraqi Army, but representatives of the Shi’ite paramilitary groups that have led the fight against Islamic State.

Hadi al-Amiri, one of the most senior militia leaders, delivered a long and forceful monologue on his fighters’ recent victories. Abadi, in a blazer and tie, listened, occasionally jotting down notes, a video of the meeting shows. A few minutes later, Abadi himself praised the fighters.

The event was a sign of the delicate power balance in Iraq. […]