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We revisit AdS_4 heterotic compactifications on nearly K\"ahler manifolds inthe presence of H-flux and certain fermion condensates. Unlike previousstudies, we do not assume the vanishing of the supersymmetry variations.Instead we determine the full equations of motion originating from theten-dimensional action, and subsequently we provide explicit solutions to themon nearly K\"ahler manifolds at first order in alpha'. The Bianchi identity isalso taken into account in order to guarantee the absence of all anomalies. Inthe presence of H-flux, which is identified with the torsion of the internalspace, as well as of fermion condensates in the gaugino and dilatino sectors,new solutions are determined. These solutions provide a full classification ofconsistent backgrounds of heterotic supergravity under our assumptions. All thenew solutions are non-supersymmetric, while previously known supersymmetricones are recovered too. Our results indicate that fully consistent(supersymmetric or not) heterotic vacua on nearly K\"ahler manifolds arescarce, even on AdS_4, and they can be completely classified.

Nearly K\"ahler heterotic compactifications with fermion condensates