Usually that is an indication that your grey tank is full. I will assume that is not the issue here.
The vents for your plumbing are in the drain lines NOT the grey water tank. They are there to insure that the liquid drains quickly and does not create a vacuum situation. I think that any water entering your grey tank is creating a pressure situation and the water bubbling up in your bathtub is from water that is in the bath tub plumbing trap. I think the only way to eliminate the problem is that you need to somehow vent your grey water tank.