The "knuckle" on the transfer case end of it failed, started twisting on itself.
And I looked at the diagram wrong - there's no guibo on the front shaft, only on the rear. But now that I've been under there a couple of times, it's become more clear that it's likely the output shaft seal that's leaking.
My problem right now is that I can't get the output shaft out so that I can replace the seal. See
https://www.e90post.com/forums/showthread.php?t=1220395.
And thanks for the advice - I'll take a close look at it when I do get it off.