I'd imagine they'd just be regular folk at this point. Or at least regular folk with an annoying habit of saying "You know, my greatx20 grandfather was Emperor".
Also, I figured out how China will fight the Spanish.
It seems like what happened. But the family name of Zhu still sticks. But at the least particulary sharp people could probably point it out and chuckle as a joke and no one would care.
What I find amusing is that our modern-day Mings have even managed to get office in modern China. There's one that served as vice-premier a fair bit before serving as the country's fifth premier.