  We construct a broad class of frustration-free quantum vertex models in 3+1D
whose ground states are weighted superpositions of classical 3D vertex model
configurations. For the simplest members of this class, the corresponding
classical vertex models have a $\mathbb{Z}_2$ gauge constraint enriched with a
$\mathbb{Z}_2$ global symmetry. We deduce a phase diagram for the quantum
vertex models using exact dualities and mappings to classical spin models and
height models. We find $\mathbb{Z}_2$ deconfined and $\mathbb{Z}_2$
symmetry-broken phases which are mapped to one another by duality. The
transition between these phases is governed by a self-dual gapless point which
is related by a different duality to the Rokhsar-Kivelson (RK) point of quantum
spin ice. Our results are illustrated for diamond and cubic lattices, but hold
for general 3D lattices with even coordination number.
