AP Photo

It is 5 a.m. in Istanbul, and I am looking for coffee. Having arrived in Istanbul's old city the night before and seriously jetlagged, I decided to walk into the Eyup quarter, which hosts Istanbul's most sacred mosque, Eyup Sultan. I hoped the revered shrine, which attracts early morning worshippers, would have an open coffee shop nearby, and I was right. As prayers ended, I watched Eyup's worshipers flow from the mosque, sipping a bland cup of instant coffee, unaware I was about to be treated to an experience of cultural flavor unique to Turkey.

A large group of Salafists, with their trademark trimmed beards and kaftans, walked out of the mosque, heading to my coffee shop. What happened next is a lesson in Turkey's distinctive direction compared to its Muslim...