  We construct examples of compact homogeneous Riemannian manifolds admitting
an invariant Bismut connection that is Ricci flat and non-flat, proving in this
way that the generalized Alekseevsky-Kimelfeld theorem does not hold. The
classification of compact homogeneous Bismut Ricci flat spaces in dimension $5$
is also provided. Moreover, we investigate compact homogeneous spaces with non
trivial third Betti number, and we point out other possible ways to construct
Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections
correspond to fixed points of the generalized Ricci flow, we discuss the
stability of some of our examples under the flow.
