My S13 does similar.

I swapped bulbs etc already.

My driver side rear will have one filament on if i have lights on and its fine. When i brake, second filament should go on as well and it does but my first one goes off. I let brake pedal go, it's fine again. Passenger side is totally fine.

I looked at the contacts and changing the 180 degree deal to no avail.

My contacts were clean and not loose.

So next is a continuity test on wires (harness to trace down a problem, probably a short somewhere)

RINKU