We consider quasigroups $Q(\Gamma)$ obtained as certain double covers of the symmetric group $S_3$ of degree 3, for directed graphs $\Gamma$ on the vertex set $S_3$. We completely characterize the strong compatibility of elements of $Q(\Gamma)$ for any quasigroup nonuniform homogeneous space of degree 4. For such homogeneous spaces, we classify all the strong and weak compatibility graphs of $Q(\Gamma)$.