  We study spin-$\frac{1}{2}$ fermions coupled to $\mathbb{Z}_2$ gauge fields
on a lattice. We show how a spatial modulation of the fermion hopping allows
for the realization of various obstructed atomic insulators that host
higher-order band topology. Studying the effect of quantum dynamics of the
gauge fields within a simplified model, we find a rich phase diagram of this
system with a number of superfluid phases arising from the attractive
interactions meditated by the gauge fields. A key finding of this work is that
the evolution from the Bardeen-Cooper-Schrieffer (BCS) superfluid state to a
Bose-Einstein condensate (BEC) of tightly bound pairs occurs via the
realization of these different superfluid phases separated by first-order
transitions.
