Each gamer would take turns being both a GM for their portion of the map, and running characters in the other player's games. That means you would have several characters coexisting at different places in the game universe. (Obviously this would require the various GMs to take turns running their games, and I don't pretend to know how that would work without conflicts.)
The game would also work on another level. You could play out the strategic conflicts of the various regions as wargaming sessions, using a space combat game or even a strategic 4X game. The results of the wargaming sessions would influence the background events of the roleplaying universe, giving the whole setting an extra level of depth.
For example, Stellar Conquest
Of course actually coordinating and running a Grand Campaign like this would be almost impossible. Maybe if you had a tight-knit gaming group that kept a regular schedule you could attempt it. But considering how hard it is just to run a an extended campaign and get everyone to show up when they're supposed to, it seems like a pipe dream. But it's a fun dream.