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

Equivariant Cartan-Eilenberg supergerbes for the Green-Schwarz  superbranes III. The wrapping anomaly and the super-${\rm  AdS}_5\times\mathbb{S}^5$ background