When we derive the exchange statistics by moving quasiparticles around a circle in the toric code model we do not mention any Berry phase contribution. Is the Berry phase contribution trivial or it is nontrivial but does not alter the exchange statistics? This is also the case when we derive the nonabelian statistics for the vortices in 2D chiral $p$-wave superconductors. We seem to consider only the wave function monodromy but not the Berry phase contribution.
My question is:
(simpler one) what is the Berry phase contribution in both cases and why does it not alter the exchange statistics?
(harder one) is there any way to reach the conclusion (i.e. trivial vs.\ nontrivial, and alter vs.\ not alter the statistics) without calculation?
(challenge) could we find a general guideline as to whether and when we should account for the Berry phase contribution when deriving exchange statistics for any topological phase?