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 $alpha$.
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