“[The Bektashi] are even said to be affiliated to some of the French Masonic Lodges. One thing is certain; the order now consists almost exclusively of gentlemen of education, belonging to the Liberal, or Young Turk party.”
- Richard Davey, The Sultan and His Subjects  (Gorgias Press LLC, 2001), p. 65.
I’m not that familiar with the Bektashi Order, but it turns out that this politico-mystical secret society - directly or indirectly - was the inspiration behind the Ancient Arabic Order of Nobles of the Mystic Shrine.
Freemasons Dr. Walter M. Fleming, 33°, and William J. Florence, 32°, founded the Shriners in 1870. And in recounting the origins of their Order, Florence and Fleming corroborate the above statement by Richard Davey.