  We consider $G_2$ structures with torsion coupled with $G_2$-instantons, on a
compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms
which appears in supergravity and generalized geometry, known as the Bianchi
identity. The resulting system of partial differential equations can be
regarded as an analogue of the Strominger system in $7$-dimensions. We initiate
the study of the moduli space of solutions and show that it is finite
dimensional using elliptic operator theory. We also relate the associated
geometric structures to generalized geometry.
