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We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on acompact $7$-dimensional manifold. The coupling is via an equation for $4$-formswhich appears in supergravity and generalized geometry, known as the Bianchiidentity. First studied by Friedrich and Ivanov, the resulting system ofpartial differential equations describes compactifications of the heteroticstring to three dimensions, and is often referred to as the $G_2$-Stromingersystem. We study the moduli space of solutions and prove that the space ofinfinitesimal deformations, modulo automorphisms, is finite dimensional. Wealso provide a new family of solutions to this system, on $T^3$-bundles over$K3$ surfaces and for infinitely many different instanton bundles, adapting aconstruction of Fu-Yau and the second named author. In particular, we exhibitthe first examples of $T$-dual solutions for this system of equations.

$T$-Dual solutions and infinitesimal moduli of the $G_2$-Strominger system