  We introduce a notion of Aubry set for weakly coupled systems of
Hamilton--Jacobi equations on the torus and characterize it as the region where
the obstruction to the existence of globally strict critical subsolutions
concentrates. As in the case of a single equation, we prove the existence of
critical subsolutions which are strict and smooth outside the Aubry set. This
allows us to derive in a simple way a comparison result among critical sub and
supersolutions with respect to their boundary data on the Aubry set, showing in
particular that the latter is a uniqueness set for the critical system. We also
highlight some rigidity phenomena taking place on the Aubry set.
