Ok, I tried a Magic Eraser to no avail, but what DID work was a handled scrub brush after soaking the area with Mean Green. Bleach and that Barkeeper's Friend stuff didn't have any meaningful affect.
There's still a bit in what looks like cracks in the finish, but the table has been in this screened porch for at least 15 years so the real fix is probably to refinish it.
It's a tad wider than I'd like, though, so I may end up building something new for the space.
Thanks for the info, all!
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