  This articles is devoted to a description of the second-order differential
geometry of torsion-free almost quaternionic skew-Hermitian manifolds, that is,
of quaternionic skew-Hermitian manifolds $(M, Q, \omega)$. We provide a
curvature characterization of such integrable geometric structures, based on
the holonomy theory of symplectic connections and we study qualitative
properties of the induced Ricci tensor. Then we proceed with bundle
constructions over such a manifold $(M, Q, \omega)$. In particular, we prove
the existence of almost hypercomplex skew-Hermitian structures on the Swann
bundle over $M$ and investigate their integrability.
