In this paper we give a necessary and sufficient condition for a (real) moment-angle complex to be a topological manifold. The cup and cap products in a real moment-angle manifold are studied. Consequently, the cohomology ring (with coefficients integers) of a polyhedral product by pairs of disks and their bounding spheres is isomorphic to that of a differential graded algebra associated to $K$ and the dimensions of the disks.
"On products in a real moment-angle manifold." J. Math. Soc. Japan 69 (2) 503 - 528, April, 2017. https://doi.org/10.2969/jmsj/06920503