Suppose $f:C\to C$ is a homeomorphism, where $C=\{0,1\}^{\mathbb N}$ is Cantor space. If $x=^* y$ implies $f(x)=^* f(y)$ (equal on all but finitely many coordinates) then is the same true for $f^{-1}$?

I know it fails without the homeorphism assumption if we use Axiom of Choice, for instance...