In several models of beyond Standard Model physics (BSM) discrete symmetries play an important role. For instance, in order to avoid flavor changing neutral currents (FCNC), a discrete $Z_2$ symmetry is imposed on Two-Higgs-Doublet-Models (2HDM). This can lead to the formation of domain walls (DW) as the $Z_2$ symmetry gets spontaneously broken during electroweak symmetry breaking (EWSB) in the early universe.
Due to this simultaneous spontaneous breaking of both the discrete symmetry and the electroweak symmetry, the vacuum manifold has the structure of two disconnected 3-spheres and the formed domain walls can exhibit several special properties in contrast to standard domain walls. We focus on some of these properties such as CP and electric charge violating vacua localized inside the domain walls. The breaking of $U(1)_{em}$ inside the wall leads to the known phenomenon of "clash-of-symmetries" mechanism, meaning that the symmetry group inside the wall is smaller than the symmetry group far from the wall. We also discuss the scattering of top quarks off such types of domain walls and show, for example, that they can be reflected or transmitted off the wall as a bottom quark.