  This is a continuation of a programme, initiated in the work arXiv:1706.05682
[hep-th], of supersymmetry-equivariant geometrisation of the Green-Schwarz
super-$(p+2)$-cocycles coupling to the topological charges carried by
super-$p$-branes of the superstring theory on reductive homogeneous spaces of
supersymmetry Lie supergroups. In the present part, the ideas and
geometro-algebraic tools developed previously are substantially enhanced,
adapted to and applied in the physically significant curved backround of
Metsaev and Tseytlin, determining the propagation of the critical superstring
in the super-${\rm AdS}_5\times\mathbb{S}^5$ geometry. The analysis brings to
the fore the r\^ole, in the geometrisation scheme proposed, of the wrapping
anomaly of the Poisson algebra of the Noether charges of the rigid symmetries
of the relevant super-{\sigma}-model that lift the geometric symmetries of the
supertarget. In particular, the significance of the charges quantifying the
monodromy of the Grassmann-odd coordinates in the Kosteleck\'y-Rabin-type
quotient of the supertarget is emphasised. A trivial super-1-gerbe is
associated with the Metsaev-Tseytlin super-3-cocycle over the super-${\rm
AdS}_5\times\mathbb{S}^5$ target. The issue of compatibility of the
geometrisation with the \.In\"on\"u-Wigner contraction of the supersymmetry Lie
superalgebra to its flat-superspace counterpart is investigated at some length,
revealing the rigidity of the relevant Cartan-Eilenberg cohomology and
signalling an attractive potential alternative to the non-contractible trivial
super-1-gerbe constructed.
