  We study the phase diagram of two different mixed-dimensional
$t-J_z-J_{\perp}$-models on the square lattice, in which the hopping amplitude
$t$ is only nonzero along the $x$-direction. In the first, bosonic, model, the
spin exchange amplitude $J_{\perp}$ is negative and isotropic along the $x$ and
$y$ directions of the lattice, and $J_z$ is isotropic and positive. The
low-energy physics is characterized by spin-charge separation: the holes hop as
free fermions in an easy-plane ferromagnetic background. In the second model,
$J_{\perp}$ is restricted to the $x$-axis while $J_z$ remains isotropic and
positive. The model is agnostic to particle statistics, and shows stripe
patterns with anti-ferromagnetic N{\'e}el order at low temperature and high
hole concentrations, in resemblance of the mixed-dimensional $t-J_z$ and $t-J$
models. At lower hole concentration, a very strong first order transition and
hysteresis loop is seen extending to a remarkably high 14(1)% hole doping.
