A pilgrim in long white cotton veils trimmed with blue embroidery emerged from the old Byzantine chapel to their right. The veils were pulled up over her head and wound around her shoulders, covering everything but a foot of shirt below the edge of the veil, almost like the way the Arab women wore their veils—only white instead of black.
Sounds like a burqa, which was and is worn in primarily by Pashtun women in Central Asia, of whom there would have been few in the Levant. Depending on what school of Islamic jurisprudence they followed, Muslim women would either have covered everything but their but their faces and hands or everything but their eyes.
I rather doubt that the concept of veiling would have seemed especially exotic to medieval Christians, though, because most Christian women above the age of puberty would have covered their hair in public, too, especially if they were married.
