Update: I can't let broken equipment sit around on my desk without at least taking a look. I opened it up and found that the input jacks have kind of a strange (I think . . . I haven't taken apart many multimeters) mounting setup. They seem to be somehow held in place in the body of the meter (normally) and then tightened against a kind of chintzy-looking soldered-on metal bar. I'm a bit surprised by how small these are--in Dave's teardown video of the U1282A, which I googled after taking this one apart, they look a lot beefier.
Anyway, whatever is supposed to keep the input port in place while the nut is tightened against it is broken on the COM port, so the whole thing turns freely when I try to tighten the nut. Whatever machine tried to tighten it before me must have had the same issue, because the nut is not tightened against the soldered bar, which moves loosely when I push on it.
I just held the threaded part of the input connector with pliers and tightened the nut and now it seems to be mostly, at least, working. The response of the meter is a bit weird when I measure zero resistance, but I think it's just that the autoranging is super slow. If I measure a 10k resistor it seems fine.
Anyway, something to check if anyone else has a similar problem with one of these. Given that there's no way this problem wasn't there since the thing was put together, I still can't imagine how it passed calibration.