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"*][1] 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.

You might be interested also in this [paper][2] by Rheinboldt. It contains some topological conditions on when the local solvability of the equation
$$F(x,f(x,z))=z$$
leads to the global solvability.


  [1]: http://books.google.co.uk/books?id=ya5yy5EPFD0C&printsec=frontcover&dq=implicit+function+theorem&cd=1#v=onepage&q&f=false
  [2]: http://www.ams.org/journals/tran/1969-138-00/S0002-9947-1969-0240644-0/home.html