  In this paper, we study a class of symmetry reduced models of $\mathcal{N}=1$
supergravity using self-dual variables. It is based on a particular Ansatz for
the gravitino field as proposed by D'Eath et al. We show that the essential
part of the constraint algebra in the classical theory closes. In particular,
the (graded) Poisson bracket between the left and right supersymmetry
constraint reproduces the Hamiltonian constraint. For the quantum theory, we
apply techniques from the manifestly supersymmetric approach to loop quantum
supergravity, which yields a graded analog of the holonomy-flux algebra and a
natural state space. We implement the remaining constraints in the quantum
theory. For a certain subclass of these models, we show explicitly that the
(graded) commutator of the supersymmetry constraints exactly reproduces the
classical Poisson relations. In particular, the trace of the commutator of left
and right supersymmetry constraints reproduces the Hamilton constraint
operator. Finally, we consider the dynamics of the theory and compare it to a
quantization using standard variables and standard minisuperspace techniques.
