Do you mean this one?
https://www.arcade-projects.com/threads/minigun-supergun-an-open-source-supergun.9408/I think Frank is active on the Aussie Arcade forums too. So might be worth looking/asking there.
Having a look over the schematics, I'd say that the sync level might be too high for consumer CRT TV. The sync is being amped identically to the RGB signals (by the THS7374 chip), which is up to 0 - 2vpp (before any termination at the TV/monitor). This would be fine for an arcade monitor, which I think is what it is designed for, but could be too high for TVs that want sync at 0 - 0.3vpp.
What you could do is to add a resistor between the device's sync output and the TV. I'd guess something like 150R - 470R. Or try using a 1K or 500R pot (only 2 legs connected) to dial it in. Use your multimeter to measure voltages before and after.
You could take the sync from TMNT direct and bypass the Minigun! If you do so, I suggest adding a ~470R+ resistor or a 1K pot as suggested above.
Also, try using your TMNT with another cab/JAMMA setup to see if it has a problem, to rule it out.
A few more details and links to confirm this is the same thing
As always, pics or it didn't happen!