  By analogy with associative and co-associative cases we introduce a class of
three-dimensional non-orientable submanifolds, of almost
$\mathrm{G}_2-$manifolds, modelled on planes lying in a special
$\mathrm{G}_2-$orbit. An application of the Cartan-K\"ahler theory shows that
some three-manifold can be presented in this way. We also classify all the
homogeneous ones in $\mathbb{RP}^7$.
