There are aspects of FPS and turn based games that make them feel like they should be mutually exclusive. Turn based games, for instance, really require teams of players, other wise the pace is a little too short, and you need the team tactical element to make up for that. Similarly, an fps gives you a limited viewport, which mitigates the range of tactical decisions.
One idea that I thought could work is if you worked integrated the fps role as part of a team. As an example:
Imagine playing the role of a long range sniper. You have, through your scope, a view of the battleground (say an enemy compound). You can see everything, the enemy soldiers, the objective, and your own team positioning themselves at the compound entry points.
The turns alternate between the sniper (you) and every one else (who is AI controlled). In this game, your turn would be looking over the battleground and choosing one target to eliminate. Once your shot has been taken, the game moves forward one step. Your team infiltrates one time unit, and the enemy responds. This would be in realtime, rather than turn based. So you act, everyone else acts.
You could even expand it with a tactical planning phase, where you set waypoints for your team on go-codes (like the original rainbow six games).
As a gross over simplification. You shoot one guard, your team infiltrates the first room, and encounters a group of enemies, the game pauses. You shoot one enemy, your team opens fire and the enemy reacts, the game pauses... and so on.
You could also incorporate some semi-fixed movement in to the game. So, instead of taking a shot this turn, you opt to move from your current position (rooftop A) to another position (rooftop B).
I think that could work pretty well...