  When studying a graded module $M$ over the Cox ring of a smooth projective
toric variety $X$, there are two standard types of resolutions commonly used to
glean information: free resolutions of $M$ and vector bundle resolutions of its
sheafification. Each approach comes with its own challenges. There is geometric
information that free resolutions fail to encode, while vector bundle
resolutions can resist study using algebraic and combinatorial techniques.
Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which
capture desirable geometric information and are also amenable to algebraic and
combinatorial study. The theory of virtual resolutions includes a notion of a
virtually Cohen--Macaulay property, though tools for assessing which modules
are virtually Cohen--Macaulay have only recently started to be developed.
  In this paper, we continue this research program in two related ways. The
first is that, when $X$ is a product of projective spaces, we produce a large
new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to
be virtually Cohen--Macaulay via explicit constructions of appropriate virtual
resolutions reflecting the underlying combinatorial structure. The second is
that, for an arbitrary smooth projective toric variety $X$, we develop
homological tools for assessing the virtual Cohen--Macaulay property. Some of
these tools give exclusionary criteria, and others are constructive methods for
producing suitably short virtual resolutions. We also use these tools to
establish relationships among the arithmetically, geometrically, and virtually
Cohen--Macaulay properties.
