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We study almost hypercomplex skew-Hermitian structures and almostquaternionic skew-Hermitian structures, as the geometric structures underlying$\mathsf{SO}^\ast(2n)$- and $\mathsf{SO}^\ast(2n)\mathsf{Sp}(1)$-structures,respectively. The corresponding intrinsic torsions were computed in theprevious article in this series, and the algebraic types of the geometries werederived, together with the minimal adapted connections (with respect to certainnormalizations conditions). Here we use these results to present the relatedfirst-order integrability conditions in terms of the algebraic types and otherconstructions. In particular, we use distinguished connections to provide amore geometric interpretation of the presented integrability conditions andhighlight some features of certain classes. The second main contribution ofthis note is the illustration of several specific types of such geometries viaa variety of examples. We use the bundle of Weyl structures and describeexamples of $\mathsf{SO}^\ast(2n)\mathsf{Sp}(1)$-structures in terms offunctorial constructions in the context of parabolic geometries.

Differential geometry of ${\mathsf{SO}}^{\ast}(2n)$-type structures -- Integrability