I will begin by recalling a standard construction of a canonical toric embedding of a (non-necessarily projective) Mori Dream Space (MDS), starting from its Cox ring. Then I will show how recent results, jointly obtained with L. Terracini for Q-factorial complete toric varieties, can be easily extended to the canonical ambient toric variety of a MDS: this construction naturally restricts to define a covering space, unramified in codimension 1, of the given MDS, which is still a MDS admitting a torsion-free class group.
In principle, an application of such a covering construction is that the Cox ring of a MDS, which is in general graduated over a class group with non trivial torsion part, could be described in terms of the Cox ring of its canonical covering, which is now graded over a torsion-free class group.
This is a work in progress.