Okay, try powering it from the 12v permanent feed and/or the 12v ignition feed instead of powering it from the amp remote line. Second, try grounding it with a proper ground instead of the serial wire's ground.

Can't come from permanent. It's a stealth box with no Off switch. I suppose it could come from ignition switched, but I'd then have to fuse it. I suspect that it is a problem within the GPS itself that needs addressing - the tuner doesn't have such problems, and current consumption should be comparable. I'd really like to keep it on RA if possible. I suppose that at a push I could rebuild the interface circuit to use RA to switch a permanent wire.

Investigation this afternoon has suggested that I'm going to need to run power and ground lines to eliminate this completely.