If the clutches are worn or the S spring is weak, one side could be slipping with your new sticky tires (that last part was for Serious).
A quick, kinda lame way to try and help it is to drain the fluid and put fresh oil in that has no friction modifier. The modifier actually helps the clutches slip, so guys who want bite (at the possible expense of chatter when one wheel is spinning faster than the other) can try this.
A rebuild kit is about 50 bucks if you want to just freshen the LSD. That is the route I would go unless this was a one-time thing for some reason.
Good luck.