Are you sure the blank tank is overflowing to the point it is back filling the toilet? Typically, that would not happen. I suspect that the toilet water valve is not completely shutting off, and thus a slight flow of water continues into the bowl until it overflows.
To confirm my suspicion, pour a bottle of food coloring into the black tank. Allow it to fill and then let the toilet overflow. If you are right, the water in the toilet bowl will be colored by the food coloring. If I am right, the water in the overflowing toilet bowl will be clear.