Suppose $f:C\to C$ is a homeomorphism, where $C=\{0,1\}^{\mathbb N}$ is Cantor space.
Suppose $f$ preserves $=^*$ (equality on all but finitely many coordinates). Does it follow that $f$ also reflects $=^*$?
That is, if
$$x=^* y \implies f(x)=^* f(y)$$
does it follow that:
$$x=^* y \iff f(x)=^* f(y)?$$

Using Axiom of Choice we can show that a bijection of $C$ that preserves $=^*$ need not reflect $=^*$, so the continuity assumptions are not superfluous.