  We summarize our geometric and topological description of compact
eight-manifolds which arise as internal spaces in ${\cal N}=1$ flux
compactifications of M-theory down to $\mathrm{AdS}_3$, under the assumption
that the internal part of the supersymmetry generator is everywhere non-chiral.
Specifying such a supersymmetric background is {\em equivalent} with giving a
certain codimension one foliation defined by a closed one-form and which
carries a leafwise $G_2$ structure, a foliation whose topology and geometry we
characterize rigorously.
