  We use the theory of singular foliations to study ${\cal N}=1$
compactifications of eleven-dimensional supergravity on eight-manifolds $M$
down to $\mathrm{AdS}_3$ spaces, allowing for the possibility that the internal
part $\xi$ of the supersymmetry generator is chiral on some locus ${\cal W}$
which does not coincide with $M$. We show that the complement $M\setminus {\cal
W}$ must be a dense open subset of $M$ and that $M$ admits a singular foliation
${\bar {\cal F}}$ endowed with a longitudinal $G_2$ structure and defined by a
closed one-form $\boldsymbol{\omega}$, whose geometry is determined by the
supersymmetry conditions. The singular leaves are those leaves which meet
${\cal W}$. When $\boldsymbol{\omega}$ is a Morse form, the chiral locus is a
finite set of points, consisting of isolated zero-dimensional leaves and of
conical singularities of seven-dimensional leaves. In that case, we describe
the topology of ${\bar {\cal F}}$ using results from Novikov theory. We also
show how this description fits in with previous formulas which were extracted
by exploiting the $\mathrm{Spin}(7)_\pm$ structures which exist on the
complement of ${\cal W}$.
