  We obtain non-trivial solutions to the heterotic $\rm{G}_2$ system, which are
defined on the total spaces of non-trivial circle bundles over Calabi--Yau
$3$-orbifolds. By adjusting the $S^1$ fibres in proportion to a power of the
string constant $\alpha'$, we obtain a cocalibrated $\rm{G}_2$-structure the
torsion of which realises an arbitrary constant (trivial) dilaton field and an
$H$-flux with nontrivial Chern--Simons defect. We find examples of connections
on the tangent bundle and a non-flat $\rm{G}_2$-instanton induced from the
horizontal Calabi--Yau metric which satisfy together the anomaly-free
condition, also known as the heterotic Bianchi identity. The connections on the
tangent bundle are $\rm{G}_2$-instantons up to higher order corrections in
$\alpha'$.

Differential Geometry;High Energy Physics - Theory