  The monolayer halides CrX$_3$ (X=Cl, Br, I) attract significant attention for
realizing 2D magnets with genuine long-range order (LRO), challenging the
Mermin-Wagner theorem. Here, we show that monolayer CrCl$_3$ has the unique
benefit of exhibiting tunable magnetic anisotropy upon applying a compressive
strain. This opens the possibility to use CrCl$_3$ for producing and studying
both ferromagnetic and antiferromagnetic 2D Ising-type LRO as well as the
Berezinskii-Kosterlitz-Thouless (BKT) regime of 2D magnetism with quasi-LRO.
Using state-of-the-art density functional theory, we explain how realistic
compressive strain could be used to tune the monolayer's magnetic properties so
that it could exhibit any of these phases. Building on large-scale quantum
Monte Carlo simulations, we compute the phase diagram of strained CrCl$_3$, as
well as the magnon spectrum with spin-wave theory. Our results highlight the
eminent suitability of monolayer CrCl$_3$ to achieve very high BKT transition
temperatures, around 50 K, due to their singular dependence on the weak
easy-plane anisotropy of the material.
