  We consider spaces of "virtual" constrained generalized Killing spinors, i.e.
spaces of Majorana spinors which correspond to "off-shell" $s$-extended
supersymmetry in compactifications of eleven-dimensional supergravity based on
eight-manifolds $M$. Such spaces naturally induce two stratifications of $M$,
called the chirality and stabilizer stratification. For the case $s=2$, we
describe the former using the canonical Whitney stratification of a
three-dimensional semi-algebraic set ${\cal R}$. We also show that the
stabilizer stratification coincides with the rank stratification of a cosmooth
generalized distribution ${\cal D}_0$ and describe it explicitly using the
Whitney stratification of a four-dimensional semi-algebraic set $\mathfrak{P}$.
The stabilizer groups along the strata are isomorphic with $\mathrm{SU}(2)$,
$\mathrm{SU}(3)$, $\mathrm{G}_2$ or $\mathrm{SU}(4)$, where $\mathrm{SU(2)}$
corresponds to the open stratum, which is generically non-empty. We also
determine the rank stratification of a larger generalized distribution ${\cal
D}$ which turns out to be integrable in the case of compactifications down to
$\mathrm{AdS}_3$.
