DUBAI—Qassem Soleimani was killed by an American strike just as Iran’s regional sphere of influence that the top military commander so successfully expanded over the past decade has come under threat—from the Middle East’s own people.

In Iraq, street protests against Tehran’s domination have raged since October, usually led by Iran’s fellow Shiite Muslims. This unrest has resulted in hundreds of deaths and involved the torching of the Iranian consulates in the Shiite holy cities of Karbala and Najaf south of Baghdad.

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