  We study stratified G-structures in ${\cal N}=2$ compactifications of
M-theory on eight-manifolds $M$ using the uplift to the auxiliary nine-manifold
${\hat M}=M\times S^1$. We show that the cosmooth generalized distribution
${\hat {\cal D}}$ on ${\hat M}$ which arises in this formalism may have
pointwise transverse or non-transverse intersection with the pull-back of the
tangent bundle of $M$, a fact which is responsible for the subtle relation
between the spinor stabilizers arising on $M$ and ${\hat M}$ and for the
complicated stratified G-structure on $M$ which we uncovered in previous work.
We give a direct explanation of the latter in terms of the former and relate
explicitly the defining forms of the $\mathrm{SU}(2)$ structure which exists on
the generic locus ${\cal U}$ of $M$ to the defining forms of the
$\mathrm{SU}(3)$ structure which exists on an open subset ${\hat {\cal U}}$ of
${\hat M}$, thus providing a dictionary between the eight- and nine-dimensional
formalisms.
