There exist several known global implicit function theorems. Those results tend to be tailored for specific applications, and it is rather difficult to state a universally useful version.
One result that I find particularly helpful goes back to Hadamard (see, e.g., Chapt. 6 of "The Implicit Function Theorem" by Krantz and Parks):
Theorem. Let $M$ and $N$ be smooth, connected manifolds of dimension $d$ and let $f:M\to N$ be a $C^1$ function. If
- $f$ is proper (i.e. $f^{-1}(K)\subset M$ is compact whenever $K\subset N$ is compact),
- the Jacobian of $f$ vanishes nowhere on $M$, and
- $N$ is simply connected,
then $f$ is a homeomorphism.