  We study noncommutative principal bundles (Hopf-Galois extensions) in the
context of coquasitriangular Hopf algebras and their monoidal category of
comodule algebras. When the total space is quasi-commutative, and thus the base
space subalgebra is central, we define the gauge group as the group of vertical
automorphisms or equivalently as the group of equivariant algebra maps. We
study Drinfeld twist (2-cocycle) deformations of Hopf-Galois extensions and
show that the gauge group of the twisted extension is isomorphic to the gauge
group of the initial extension. In particular, noncommutative principal bundles
arising via twist deformation of commutative principal bundles have classical
gauge group. We illustrate the theory with a few examples.
