  We introduce an algebra Z[X,S] associated to a pair (X,S) of a virtual birack
X and X-shadow S. We use modules over Z[X,S] to define enhancements of the
virtual birack shadow counting invariant, extending the birack shadow module
invariants to virtual case. We repeat this construction for the twisted virtual
case. As applications, we show that the new invariants can detect orientation
reversal and are not determined by the knot group, the Arrow polynomial and the
Miyazawa polynomial, and that the twisted version is not determined by the
twisted Jones polynomial.
