  At variance from the cases of relative equilibria and relative periodic
orbits of dynamical systems with symmetry, the dynamics in relative
quasi-periodic tori (namely, subsets of the phase space that project to an
invariant torus of the reduced system on which the flow is quasi-periodic) is
not yet completely understood. Even in the simplest situation of a free action
of a compact and abelian connected group, the dynamics in a relative
quasi-periodic torus is not necessarily quasi-periodic. It is known that
quasi-periodicity of the unreduced dynamics is related to the reducibility of
the reconstruction equation, and sufficient conditions for it are virtually
known only in a perturbation context. We provide a different, though
equivalent, approach to this subject, based on the hypothesis of the existence
of commuting, group-invariant lifts of a set of generators of the reduced
torus. Under this hypothesis, which is shown to be equivalent to the
reducibility of the reconstruction equation, we give a complete description of
the structure of the relative quasi-periodic torus, which is a principal torus
bundle whose fibers are tori of a dimension which exceeds that of the reduced
torus by at most the rank of the group. The construction can always be done in
such a way that these tori have minimal dimension and carry ergodic flow.
