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 neat 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. Furthermore, we show that the trace of any critical subsolution on this set can be extended to the whole torus in such a way that the output is a critical solution. We also highlight some rigidity phenomena taking place on the Aubry set: first, the values taken by the differences of the components of a critical subsolution, on this set, are independent of the specific subsolution chosen; second, for each point $y$ in the Aubry set, there exists a vector which is a reachable gradient at $y$ of any critical subsolution.