  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. First studied by Friedrich and Ivanov, the resulting system of
partial differential equations describes compactifications of the heterotic
string to three dimensions, and is often referred to as the $G_2$-Strominger
system. We study the moduli space of solutions and prove that the space of
infinitesimal deformations, modulo automorphisms, is finite dimensional. We
also provide a new family of solutions to this system, on $T^3$-bundles over
$K3$ surfaces and for infinitely many different instanton bundles, adapting a
construction of Fu-Yau and the second named author. In particular, we exhibit
the first examples of $T$-dual solutions for this system of equations.
