Is everything running off the same breaker circuit? Ground levels among the AC circuits in your house can vary.
Did you check this? The easiest way to be sure everything's running on the same circuit is to use a power strip to get everything running off the same outlet. This will eliminate any problem here, it's worth ruling it out just to be sure.
...earlier today it was running fine without issues. Then, I had to move things around (so disconnect the cable) and now it's starting to happen again (although milder than before)...
If it was working perfectly and then you moved it and it got messed up, it possible that there's just a loose connection somewhere. Rule that out next. Double check all your connections, make sure everything's inserted all the way, make sure there's no loose components on the GBS board or the NEO GEO board making intermittent contact. While you're at it check for swelling or leaking capacitors, anything that looks weird on the boards, etc.
It is getting RGB sync from a NEO GEO MVS board...
There's no such thing as "RGB" sync. RGB just refers to three separate lines containing red, blue, and green color space information, and does not specify what kind of sync is being used. Horizontal and vertical sync can be transmitted on two more lines. RGB plus separate H and V is called RGBHV, this is what VGA is. Instead of separate H and V, composite sync (H and V combined) can be carried on one line. RGB plus composite sync is called RGBS. Sometimes composite sync can be added to the green line of RGB instead of using its own line, called sync on green. Composite video can also be used on its own line as sync.
I'm guessing that your NEO GEO is outputting composite sync on its own line, which the GBS should work fine with. Its video comes out on five lines right (RGB, sync, and ground)? How are you physically connecting the video from the NEO GEO to the GBS?
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My other thought is that I should take back my statement about the 700W power supply being fine. I've been looking at powering things with ATX power supplies, and apparently bigger is not always better. The way these things are designed, they can have problems when you're running them on a load of less than 15% of their max rating, which you probably are. I would look at getting a much smaller one, like what lilshawn suggested.
Didn't have an ATA connector, so I've improvised pins from a couple of nails taped to the wires.
Must not have noticed this before. This is not the way to troubleshoot. You can't isolate the problem when you're adding more problems. Taping things is not a consistent connection. If you didn't have a connector you should have just cut and soldered. I wouldn't worry about since you should probably stop using that power supply anyway, but make sure you're making better connections than this in the future.
...could that be a symptom of the sync voltage being too high?
Yeah maybe. As I said, it's possible it's too high or too low.
Get everything on the same circuit.
Check all your connections.
Let me know how you're getting video from the NEO GEO to the GBS.
Try combinations of different power supplies and lowering the sync level with resistors to see if it gets better or worse.
Have you ever had the NEO GEO board working on anything else, what if it has a problem itself?
What kind of monitor/tv are you using?
That manual is pretty useless, try finding someone on the shmups forum who knows more about these things.