As the time for afternoon prayers approaches, Onder Soy puts on a white robe and cap and switches on the microphone in a small 19th century room adjoining the Hagia Sophia. Soon, Soy’s melodic call to prayer rings out over a square filled with tourists hurrying to visit some of Turkey’s most famous historical sights before they close for the day. The room Soy is in — built as a resting place for the sultan and now officially called the Hagia Sophia mosque — fills up with around 40 worshipers, drawn not by the modestly decorated space itself, but by...