It's a GE TBX25PASJRAA. Water collects in the refrigerator compartment, a tray placed at the back of the top shelf catches most of it.
I assume it's defrost water that's not going where it's supposed to. I had originally thought the drain line was plugged but when I pulled the fridge out I see it contains a trap with water in it which strongly implies it's not blocked. This is part #315 on this diagram:
Also, I'm not finding any manuals that indicate how to work on it. Do such things exist anymore?