House Thuranni is a split off faction of House P. They are, unlike the other houses, direct competitors.
If we were to intoduce House T into the game as an actual house, we could then introduce direct, non-combat, pvp into the game.
What do I mean by Direct, non-combat, pvp?
I mean that party's directly compete inside of a mission, but do so in ways that do not result in direct combat between the two parties.
This could be a race between two members of the opposing parties, while the other team members activate traps, release monsters, and other actions that indirectly stop the opposing team's runner.
The key though is that the conflict does not erupt into normal player fighting normal player, causing all sorts of balance issues that need to be tweaked because of it.