I *think* the viscous coupler survives even though it is working overtime. Although the fact that the car even moves with a broke rear axles means the VC is really taking a beating. The thing that kills the tranny is due to the fact that the center diff is open. The transmission tries to send all the power to the rear wheels because there is no load*. Imagine a FWD car with an open diff. One wheel on ice, the other on dry road. The wheel on ice is gonna spin like hell. In this case, with a rear axle broke, it's as if both rear wheels are on ice, and the fronts are on dry road. So what happens is the driveshaft that goes to the rear diff is going to be spinning WAY faster than it should be trying to send power there. The viscous coupler fights this, and is still trying to send power to the front wheels. As this is happening the spider gears in the center diff are spinning much faster than normal because one output spider is essentially stationary (the front diff) and the other is moving (the rear diff/drivershaft). Normally the spider gears are not moving relative to each other. They only start to spin at all when you are turning the car and the front and rear wheels have different rates of total rotation, and even then it isn't that much. Same goes for your front and rear diff, the spider gears are stationary relative to each other until you make a turn. So the spider gears spinning crazy fast in the center diff start to EAT AWAY the center diff housing and the spider gears where they contact the housing. This is because they have no bearings. They don't have any because you don't need them normally. So all this ground up metal gets puked into your transmission chuttering up the whole freakin thing.
My buddy did this to his car and when he drained his tranny it looked like liquid silver. No way a tranny is gonna be in good shape with all those tiny steel particles floating around.
*The reason there is no load transmitted to the rear wheels is because with one half shaft busted, the rear diff will spin AROUND the spider gear that goes to the good axle. An open diff with one axle busted cannot mechanically transfer load to the other axle. You could also wind up with the same situation of spider gears eating away at the diff housing like stated above. Although it should be even worse on the rear diff because there is no viscous coupler still trying to maintain relatively equal speeds. We never figured out if this was the case with my buddy's rear diff, I suspect it was even worse. However he replaced the entire thing with a 4-bolt because the broken portion of the half-shaft was stuck in the rear diff.
ANOTHER BUSTED REAR AXLE TEST:
This one is really easy. Jack up only the rear of the car, leave the front wheels on the ground. Put the transmission in gear. Rotate one of the rear wheels by hand. If the other wheel spins the same speed in the opposite direction, your axles are fine. If the other wheel doesn't rotate, or at a much slower speed, an axle is busted. Double deluxe test: Have someone hold one of the rear wheels, you try to rotate the other one. If the other person can hold the other wheel still and you can rotate yours, an axle is busted.