  The singular values of products of standard complex Gaussian random matrices,
or sub-blocks of Haar distributed unitary matrices, have the property that
their probability distribution has an explicit, structured form referred to as
a polynomial ensemble. It is furthermore the case that the corresponding
bi-orthogonal system can be determined in terms of Meijer G-functions, and the
correlation kernel given as an explicit double contour integral. It has
recently been shown that the Hermitised product $X_M \cdots X_2 X_1A X_1^T
X_2^T \cdots X_M^T$, where each $X_i$ is a standard real complex Gaussian
matrix, and $A$ is real anti-symmetric shares exhibits analogous properties.
Here we use the theory of spherical functions and transforms to present a
theory which, for even dimensions, includes these properties of the latter
product as a special case. As an example we show that the theory also allows
for a treatment of this class of Hermitised product when the $X_i$ are chosen
as sub-blocks of Haar distributed real orthogonal matrices.
