My BMW 325ix has a center diff that occasionally behaves like you described - the engine lugs down a bit in tight corners. It has always had matched tires (always bought sets of four, never had a flat, etc, etc) so while that may cause such behavior it's not a requirement.
Note that differentials can fail in different ways, so there are multiple unrelated correct answers to your question. In your case and mine, I think the differentials (strictly speaking) are working just fine, but the viscous couplings are failing.
I wouldn't be at all surprised if VCs also had a failure mode that makes them irrelevant, leaving the differential effectively just an open diff.