  Using extensive Monte Carlo simulations, we investigate the critical
properties of domain walls, vortices and $\mathbb{Z}_2$ vortices in the
Ising-$O(2)$ and Ising-$O(3)\otimes O(2)$ models. We have consider the
nontrivial case when disorder in the Ising order parameter induces disorder in
the continuous parameter. Such a situation arises when a domain wall becomes
opaque for continuous parameter correlations. We find that in this case the
vortex density at the BKT transition (or crossover) point turns out to be
non-universal, while the wall density at the Ising transition remains
universal, i.e. in agreement with the Ising model. An important part of this
study is the numerical measurement of defect-defect correlators. We find that
the wall-vortex correlator tends to zero in the thermodynamic limit at the
Ising point, which explains the universality of the wall density. A possible
multicritical behavior of the models is also discussed.
