Emboldened as they mop up the last Islamic State forces in the city of Tikrit, Iraqi military leaders are already vowing to follow up that operation with a much more ambitious one: marching into the vast Sunni heartland in western Iraq to root out some of the most significant militant strongholds.

Iraqi and American officials say some progress in that region, Anbar Province, will be necessary before a serious effort is mounted to retake the northern city of Mosul from the Islamic State, also known as ISIS or ISIL.

But just how that can be accomplished is a source of concern. Despite boasts by the Shiite militias that they were ready for Anbar, some Iraqi and American officials say it would be disastrous for a mostly Shiite-led force to begin assaulting towns in the Sunni-dominated area, and are seeking to limit the militias’ involvement. […]