This is an old topic, but just in case anyone else is searching for this in the future, you can use the javascript replaceState to change the history and browser bar label. A simple php function to do this:
function set_url( $url )
{
echo("<script>history.replaceState({},'','$url');</script>");
}
Then would simply call this function with the desired url (presumably dropping the post variables):
set_url("http://example.com");
A page reload or a back after calling another page will now have the new url location in the history.
I think that using POST may be a more elegant solution, but if you must use GET this is a work around.