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From what I read about culture of Tuareg people of North Africa, unlike what may be stereotypically expected from a "North African/Middle Eastern" culture, it is Tuareg men rather than Tuareg women who are expected to veil their face in presence of strangers.
So the challenge is to come up with a way for a "men must cover their face in public" tradition to become widespread in large part of the world. It doesn't have to involve Tuareg or Middle East if you don't want to.