What are the exact factors that determine what faction you can become affiliated with?

I use the rephack list and other threads about how to fix up my reputation with other factions, but really wondering what's needed to get the faction tag on your name. When you start the game it's going to be "Navy_".

The only indication on how that happens is that maybe because I start very green to them.. but also am very green to a few other factions (Liberty Police and LSF I believe).

So let's say I want to get a "Zoner_" faction, would I have to be green with only the zoners and no one else? Or for that matter how about getting a "Corsair_" one? Just green with corsairs and not green with outcasts?

Thanks.