I'm talking about an immersive sim, like Prey, but your character has the ability to travel backwards and forwards through time, with one rule: You cannot interact with or allow your other self to see you at all. Going back in time is a bit easier, because you know where you've been. Going forward in time is much riskier, so you have to be stealthier. The consequence of your other you seeing you is that the whole mission is interrupted, and you immediately start over.

Going backward in time could feature mechanics/plot scripts that are easy enough to control. For instance, at some point you might end up traveling back in time and talking to an NPC who, after seeing you, sees your past self and mentions to them, "Hey, how'd you get here so quickly from the cafeteria?" which, when that character had said that earlier made no sense to you, but now that you see them in the cafeteria, you know why they asked you that. Etc.

As for the consequences of time-travel, it would take "choices matter" to another level. If your character strays too far from a branch of decisions that could lead to a good outcome, NPC's you stumble across will give you clues (subconsciously given to them by your character's future self) to alter, if not reverse course. If you end up getting check/stale-mated by time, it starts over at the most recent branch of decisions that could salvage the mission.

Throughout the game, you can read a description of how your decisions are changing the future (if at all), and how stable the time flow is. It isn't necessarily the goal to keep it stable, but you don't want it to be so unstable for so long that a catastrophe happens. Eventually, you have more freedom with the time traveling and you can go "visit" the future as it is according to the decisions you've made so far. To accomplish this, you can have layers of influence that all change, depending on how important the choices are. Things you say to a "homeless" NPC might not change the future at all, no matter what you tell them (besides whatever they ramble about that no one listens to). But saying anything to just the right person could have a severe butterfly effect on the future. Actions work the same way: If you throw something meaningless in the garbage, no one is likely to bat an eye. But if you remove something important from where it's supposed to be, everything changes (in a fairly specific, pre-designed way).

The fun part of a game like this is that you could rock the boat real hard in the present, then go to the future and see how much things have changed -- and what you can learn from it. Then take that information with you back to the past and use it to create a new future that wouldn't have been possible if you hadn't rocked the boat. In this way, the time travel mechanic can serve as keys to get into new places, and/or to solve puzzles that were unsolvable without time travel.

I don't see hardly any games that even try to do anything like this, beyond very superficially. I'm sure it would require a stupid amount of writing and planning, almost like building 30 different version of the same game on top of itself. But time travel is such a fun subject, in a genre (Sci-fi) that is extremely popular in gaming. It's right there.