  We introduce the coupled instanton equations for a metric, a spinor, a
three-form, and a connection on a bundle, over a spin manifold. Special
solutions in dimensions $6$ and $7$ arise, respectively, from the
Hull--Strominger and the heterotic $\operatorname{G}_2$ system. The equations
are motivated by recent developments in theoretical physics and can be recast
using generalized geometry; we investigate how coupled instantons relate to
generalized Ricci-flat metrics and also to Killing spinors on a Courant
algebroid. We present two open questions regarding how these different
geometric conditions are intertwined, for which a positive answer is expected
from recent developments in the physics literature by De la Ossa, Larfors and
Svanes, and in the mathematics literature on Calabi--Yau manifolds, in recent
work by the second-named author with Gonz\'alez Molina. We give a complete
solution to the first of these problems, providing a new method for the
construction of instantons in arbitrary dimensions. For
$\operatorname{G}_2$-structures with torsion coupled to
$\operatorname{G}_2$-instantons, in dimension $7$, we also establish results
around the second problem. The last part of the present work carefully studies
the approximate solutions to the heterotic $\operatorname{G}_2$-system
constructed by the third and fourth authors on contact Calabi--Yau
$7$-manifolds, for which we prove the existence of approximate coupled
$\operatorname{G}_2$-instantons and generalized Ricci-flat metrics.
