Eleven years after U.S. forces launched what’s widely described as the fiercest American military operation since the Vietnam War, Iraqi forces loyal to the government in Baghdad have begun a long-awaited operation to retake the predominantly Sunni Muslim city of Fallujah.

When the Americans set out to capture Fallujah in April 2004, it was under the control of al-Qaida in Iraq, the precursor of the Islamic State, whose fighters have held onto the city for the past 18 months.

This time, the Americans plan only a peripheral role in the coming fight. In the lead now are Iranian-trained and -equipped Shiite Muslim militias — some of whose members, ironically, fought against American forces during the U.S. occupation of their country. […]