  We consider N=1, d=4 vacua of heterotic theories in the large radius limit in
which alpha' << 1. We construct a real differential operator $\mathcal{D}=
D+\bar{D}$ on an extension bundle $(Q, \mathcal{D})$ with underlying topology
$Q=(T^{1,0}X)^* \oplus {\rm End} \, E \oplus T^{1,0} X$ whose curvature is
holomorphic and Hermitian-Yang-Mills with respect to the complex structure and
metric on the underlying non-Kahler complex 3-fold X if and only if the
heterotic supersymmetry equations and Bianchi identity are satisfied. This is
suggestive of an analogue of the Donaldson--Uhlenbeck--Yau correspondence for
heterotic vacua of this type.
