tl;dr Players can and should be allowed to join any NPC corp, and it won't break Eve.
Since the release of Faction Warfare, CCP has proven that player and even entire corps can fly under the banner of an NPC corp of their choosing, and that this has no serious consequences to the IP, the server, or any other part of the game.
And so I propose that this should be expanded, to allow players and player corporations to join virtually all NPC organizations, with perhaps the exception of CONCORD and Jovian corps. This has been a desire of almost every role-player within the game for quite a while, and could lead to greater immersion on the part of the larger Eve community.
There have long been a number of player corporations and even alliances which identify themselves with one or more NPC corporations, either as part of serious-business RP, or just for laughs. Some, such as VETO and Ghost Festival (Guristas and Angels, respectively) work on reputation alone. Others, such has Lai Dai Infinity Services and the Caldari Independent Naval Reserves, have actually made their chosen NPC organization a part of their name. This sort of immersion should be encouraged, as it helps integrate player organizations with the Eve IP in a subtle yet indelible way.
This could be expanded even further of course, allowing a slight acceleration of Loyalty Points for players that do missions for their chosen corps, or special
corp-vs-corp mission chains in proper cyber-punk tradition, but these would just be add-ons really... the Name is the thing.
Though one potential drawback might be that Live Events characters would not be so instantly recognizable when posting on the Intergalactic Summit, this could be alleviated by merely displaying a stylized star or some other symbol within the post header, to signify that they are indeed a CCP controlled character. Overall, everyone would react to them in the same enthusiastic fashion as they already do..
Finally, it shouldn't be very hard to do:
After consulting with Chribba and some other veteran players in my research, I found that this was already the case at one point, with players able to join any corporation in the game, even Concord. This didn't protect them from being Concordokken'd at all, but they were able to do it. At some point though, CCP decided to change this, and so everyone not in a player corporation was shunted into one of a few default corps, such as Perkone or The Scope. Changing this back won't likely require much effort, and might even be considered among the 'low hanging fruit' category of CSM requests.