I feel so lucky now to have this odd-looking thing....not! Thanks for the ideas/comments. I think it probably is a "plant" based thing- fungus or such. Hopefully Oscar's people will come up with an ID, but until then, I think I will try a fungicide spray.
What ever they are, they seem to have good taste- they are on my grafted white sapote but not on my crappy seedling one that is right next to it.