Abir Sabri, celebrated for her alabaster skin, ebony hair, pouting lips and full figure, used to star in racy Egyptian TV shows and movies. Then, at the peak of her career a few years ago, she disappeared—at least her face did. She began performing on Saudi-owned religious TV channels, with her face covered, chanting verses from the Qur'an. Conservative Saudi Arabian financiers promised her plenty of work, she says, as long as she cleaned up her act. "It's the Wahhabi investors," she says, referring to the strict form of Sunni Islam prevalent in Saudi Arabia. "Before, they invested in terrorism—and...