BAGHDAD - Fatima Ali was a 24-year-old divorcee with no high school diploma and no job. Shawket al-Rubae was a 34-year-old Shiite sheik with a pregnant wife who, he said, could not have sex with him. Ali wanted someone to take care of her. Rubae wanted a companion. They met one afternoon in May at the house he shares with his wife, in the room where he accepts visitors seeking his religious counsel. He had a proposal. Would Ali be his temporary wife? He would pay her 5,000 Iraqi dinars upfront — about $4 — in addition to her monthly...