KAIFENG, China—Zhang Xinwang, a moon-faced Chinese man with a spiky beard, calls himself "Moishe."

"So do you think I look Jewish?" he asks.

For much of the past millennium, Jews in Kaifeng— descendants of merchants who arrived here from Persia, probably around the 11th century—have been struggling with an existential question: What does it mean to be Jewish?

The handful of Kaifengers who go to Israel are sometimes floored to discover they need to go through a rabbi-certified conversion to be accepted as Jews, while the ones staying home squabble over which of them are really Jewish.

The question has surprising consequences in this dusty walled city in central China. According to the Chinese government, there are no Kaifeng Jews because there are no Chinese Jews. Judaism isn't one of China's five official religions and Jews aren't designated as one of the country's 55 official minorities. Orthodox Jews have a similar view, though for different reasons. Kaifeng Jews trace their heritage through their father, as Chinese traditionally do, while orthodox Jews define Judaism as passing through the mother.