I will be as thorough as possible. Both breakers will stay on. If I flip a light switch in the bathroom the gfci on breaker A will trip but breaker B stays on. Now if I reset GFCI while the light is still on it will stay on. Until I turn the light off and then on again. However IF I have more lights on breaker B on then it trips less frequently. But none the less it WILL trip. I have a family moving in this weekend to this rental and want to make sure they are going to be safe or not have to be bothered by this?
When a GFCI trips, people often say "I don't understand. Why is my GFCI tripping?"
GFCI's detect ground faults. It's what they do.
Yes, sometimes GFCIs are just bad. But Occam's Razor says the simplest explanation is usually the correct one: There's a ground fault.
Electricity flows in loops. Every hot wire has a partner neutral dedicated to it, and it's important they not be mixed. All the power that goes out on a hot, should come back via the partner neutral. If current goes any other way, that's what GFCI's call a ground fault. GFCI's compare the current on both wires, and make sure they are equal. They do not look at the ground wire at all. Ground usually figures into it as the alternate circuit path.
A hot-ground fault will usually flow a LOT of current and trip the circuit breaker. But a neutral-ground fault is trickier. They're at nearly the same voltage, but if they are brushing against each other, current will return via both neutral and ground, and cause the inequality that results in a GFCI trip.
When two separate circuits have related GFCI trips, it's usually because someone got sloppy and mixed neutrals or grounds. For instance if power flowed out one circuit's hot and the other circuit's neutral, that should trip both GFCI's. If a ground wire is being bootlegged as a neutral or is shorting against a neutral, that will also cause misrouting of power and a GFCI trip. Or if something is getting wet, it may cause enough leakage to trip a GFCI.