  We study the phase diagram and critical behaviors of three-dimensional
lattice ${\mathbb Z}_2$-gauge $N$-vector models, in which an $N$-component real
field is minimally coupled with a ${\mathbb
  Z}_2$-gauge link variables. These models are invariant under global O($N$)
and local ${\mathbb Z}_2$ transformations. They present three phases
characterized by the spontaneous breaking of the global O($N$) symmetry and by
the different topological properties of the ${\mathbb
  Z}_2$-gauge correlations. We address the nature of the three transition lines
separating the three phases. The theoretical predictions are supported by
numerical finite-size scaling analyses of Monte Carlo data for the $N=2$ model.
In this case, continuous transitions can be observed along both transition
lines where the spins order, in the regime of small and large inverse gauge
coupling $K$. Even though these continuous transitions belong to the same $XY$
universality class, their critical modes turn out to be different. When the
gauge variables are disordered (small $K$), the relevant order-parameter field
is a gauge-invariant bilinear combination of the vector field. On the other
hand, when the gauge variables are ordered (large $K$), the order-parameter
field is the gauge-dependent $N$-vector field, whose critical behavior can only
be probed by using a stochastic gauge fixing that reduces the gauge freedom.
