The first time a band of Islamic State militants “visited” the monks, they presented the monks with a kind of suggestion, in a nonthreatening manner: “Why don’t you become a Muslim?” Already, the four monks at the ancient Syriac Catholic Mar Behnam Monastery in Khidr, Iraq, had felt they were under siege. Ten days earlier, on June 10, 2014, five carloads of militants roared through the peaceful road leading to Mar Behnam, announcing through megaphones that the Islamic State was in control. Not long before that, the Iraqi army had withdrawn from a checkpoint near the monastery, located southeast of...