  Let $\mathcal{M}_g$ be the moduli space of genus $g$ Riemann surfaces. We
show that an algebraic subvariety of $\mathcal{M}_g$ is coarsely dense with
respect to the Teichm\"uller metric (or Thurston metric) if and only if it is
all of $\mathcal{M}_g$. We apply this to projections of
$\operatorname{GL}_2(\mathbb{R})$-orbit closures in the space of abelian
differentials. Moreover, we determine which strata of abelian differentials
have coarsely dense projection to $\mathcal{M}_g$.
