  A concrete geometrisation scheme is proposed for the Green-Schwarz cocycles
in the supersymmetric de Rham cohomology of the super-minkowskian spacetime,
which determine standard super-$\sigma$-model dynamics of super-p-branes. The
scheme yields higher-supergeometric structures with supersymmetry akin to those
known from the un-graded setting - distinguished (Murray-type) p-gerbe objects
in the category of Lie supergroups. These are shown to carry a canonical
equivariant structure for the action of the discrete Kosteleck\'y-Rabin
subgroup of the target supersymmetry group on the target, and thus to resolve
the `topology' of the corresponding Rabin-Crane soul superorbifold of the
super-minkowskian spacetime. The equivariant structure is seen to effectively
define a novel $\sigma$-model of super-p-brane dynamics in the super-orbifold
through the construction of the corresponding (gauge-)symmetry defect.
