Yes. You are referring to a locking differential, of which there are several types. Some use clutch plates, and will work even if one wheel has no traction at all, others use more complex gearing and can transfer torque from one to another at some multiple (called a Torsen Differential) and cannot drive when one wheel is completely unloaded.
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Is it possible to create a "negative differential gear" that puts more torque into the wheel that is stuck? (in the hopes of forcing it unstuck)
Yes. You are referring to a locking differential, of which there are several types. Some use clutch plates, and will work even if one wheel has no traction at all, others use more complex gearing and can transfer torque from one to another at some multiple (called a Torsen Differential) and cannot drive when one wheel is completely unloaded.
Awesome, thank you!