  Eleven-dimensional supergravity can be formulated in superspaces locally of
the form $\mathbf X\times Y$ where $\mathbf X$ is 4D $N=1$ conformal superspace
and $Y$ is an arbitrary 7-manifold admitting a $G_2$-structure. The
eleven-dimensional 3-form and the stable 3-form on $Y$ define the lowest
component of a gauge superfield on $\mathbf X \times Y$ that is chiral as a
superfield on $\mathbf X$. This chiral field is part of a tensor hierarchy
giving rise to a superspace Chern-Simons action and its real field strength
defines a lifting of the Hitchin functional on $Y$ to the $G_2$ superspace
$\mathbf X\times Y$. These terms are those of lowest order in a superspace
Noether expansion in seven $N=1$ conformal gravitino superfields $\Psi$. In
this paper, we compute the $O(\Psi)$ action to all orders in the remaining
fields. The eleven-dimensional origin of the resulting non-linear structures is
parameterized by the choice of a complex spinor on $Y$ encoding the off-shell
4D $N=1$ subalgebra of the eleven-dimensional super-Poincare algebra.
