  We classify 7-dimensional cocalibrated $\G_2$-manifolds with parallel
characteristic torsion and non-abelian holonomy. All these spaces admit a
metric connection $\nabla^{\mathrm{c}}$ with totally skew-symmetric torsion and
a spinor field $\Psi_1$ solving the equations in the common sector of type II
superstring theory. There exist $\G_2$-structures with parallel characteristic
torsion that are not naturally reductive.
