  The monopole-dimer model is a signed variant of the monomer-dimer model which
has determinantal structure. We extend the monopole-dimer model for planar
graphs (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof and show
that the partition function of this model can be expressed as a determinant of
a generalised signed adjacency matrix. We then show that the partition function
is independent of the orientations of the planar graphs so long as the
orientations are Pfaffian. When these planar graphs are bipartite, we show that
the computation of the partition function becomes especially simple. We then
give an explicit product formula for the partition function of
three-dimensional grid graphs a la Kasteleyn and Temperley--Fischer, which
turns out to be fourth power of a polynomial when all grid lengths are even.
Finally, we generalise this product formula to $d$ dimensions, again obtaining
an explicit product formula. We conclude with a discussion on asymptotic
formulas for the free energy and monopole densities.
