Different convergence requirement (relative tolerance) for different components of coupled model

I am working on a time-dependent model that couples two different physics branches. If I disable the second branch in the time-dependent solver, I can get a solution to converge with a relative tolerance of 0.1. If I disable the first branch (and enable the second), I need to use a much larger relative tolerance (~1000) to get the solution to converge.

Furthermore, I do not get a convergent solution in the first branch if I use this larger relative tolerance.

Is there a way to impose different tolerances for the different variables that I am solving for?

Sorry for the lack of details. Hopefully my question is general enough that they are unnecessary.

Thank you.