  We study almost hypercomplex skew-Hermitian structures and almost
quaternionic 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 the
previous article in this series, and the algebraic types of the geometries were
derived, together with the minimal adapted connections (with respect to certain
normalizations conditions). Here we use these results to present the related
first-order integrability conditions in terms of the algebraic types and other
constructions. In particular, we use distinguished connections to provide a
more geometric interpretation of the presented integrability conditions and
highlight some features of certain classes. The second main contribution of
this note is the illustration of several specific types of such geometries via
a variety of examples. We use the bundle of Weyl structures and describe
examples of $\mathsf{SO}^\ast(2n)\mathsf{Sp}(1)$-structures in terms of
functorial constructions in the context of parabolic geometries.
