Very carefully, and with a plan that does not require the players to take specific actions.
My experiences aren't exactly suited to this situation. But I think that this in-game secret is not fundamentally different from any other in-game secret. I have run politically-themed games filled with secret plans, double crosses, deceptions, and the like. My players were expected to profit from figuring out the truth, and to run into complications where they were not able to do so, and I invested a lot of effort in making sure those opportunities were fair to them. A few things that I learned:
1. You need a thorough plan.
100% required. The major danger is that, because a game master must describe everything the PCs learn about the world, and it's impractical to describe every minute detail about the world generally, a GM will only describe pertinent information.
That's a problem for secrets the players are meant to discover, because if you only describe the clues that reveal the answer to the mystery they can stand out too much and make the central mystery obvious. But it's also very easy to go too far in the other direction and hide the clues among red herrings and irrelevant details that it's too difficult for players to get and interpret the necessary information correctly.
In this scenario you can do that by planning out, in advance, what potential signs indicate a "real" reality versus a fake "reality". These should be consistent (sign A always suggests reality or unreality, and never changes) but not definitive (sign A should never be enough, on its own, to conclude anything about reality or unreality). A combination of all clues (or some minimum number, as determined by you) should be enough to clearly answer the question.
Before the first session you should have in your mind a comprehensive list of clues, real and misleading, that your players will encounter. These can be specific things (a symbol that represents a fake reality) or types of things (like a physical property of the fake reality not quite matching with true reality, causing oddities that might be observed at any time). But you should always be ready to describe a clue correctly in its context, because missing one makes the game arbitrarily more difficult and frustrating for your players.
My suggestion based on my personal experience is that pre-planned clues work the best. It's too easy to make a mistake while improvising a dozen other details, and if the plan starts to go sideways it's very difficult to wrench it back on track without giving away a lot of secret information. It's also just very difficult to consistently and fairly provide subtle clues while improvising.
2. You should be extremely disciplined in how you describe things in your game.
As above, it's no good to undermine the mystery with metagame actions. If there is a location with a clue, you should describe that location in roughly the same way each time. That's not to say that nothing can change, but if you first describe an office as containing a desk, a chair, a bookshelf, and an electric fan, it's important to account for each of those on subsequent descriptions. Even changing the order in which you describe things can potentially give hints to your players.
If the elements you describe change each time, except for the bookshelf, a player may determine that the bookshelf deserves some extra attention as a potential clue. If you're intending that result then there's no problem. But if it's inadvertent then you're removing a lot of the subtlety from your hint.
It's challenging, but in this sort of game the onus is on you to present a reliably consistent setting in which your players' logical deductions are the deciding factor in their success or failure.
3. It's important to bring clues to the players or bring players to clues. Players should not be expected to come across and identify a minimum necessary number of clues all on their own.
Since the major element of this setup is that players are uncertain of what's real and what's not it will be difficult for them to knowingly go to the right places and do the right things that will cause clues to become observable. It's already difficult to get players to do the "right" things in a less-flexible story, and efforts to make your clues subtle will exacerbate that problem.
You don't have to highlight every clue (or any of them), but it is important that you make sure your players have had the opportunity to perceive and interpret enough clues (if not necessarily all of them) to have a realistic chance of reaching the correct conclusions.
4. Players should not get conclusive feedback on their conjectures before you're ready for the mystery to be revealed.
Player cleverness in figuring things out should usually give them some kind of edge, but in this case the climax of the entire game is based around them figuring out the truth. As a consequence of that players should get no feedback of any kind on their theories, guesses, or conclusions until that point. They can check their assumptions by coming up with an idea, looking for similar clues, and seeing if their predictions are correct, and then feel stronger in their reasoning.
But ideally you would be totally blank-faced throughout. They should not be able to get information on the central mystery beyond what their characters can directly perceive and their own reasoning. No matter how sound their tentative conclusions are, they should always be concerned that they don't have complete information and could be mistaken.
5. The clues should not be obvious, individually, but should be suggestive when considered together.
This is standard fare for mystery stories, but is worth mentioning here explicitly. It adds a lot to the mystery-solving experience for players to notice clues on their own, recognize that they might matter, and then follow up of their own volition to examine them and think about their significance.
This means working out in advance what sorts of clues you want to be available (as in (1), above), and then adding unrelated descriptions of similar things so that the clues don't stand out. Considering all of the clues you want to leave in advance lets you adjust those red herring descriptions so that the clues are more or less consistently hidden throughout. This applies both to the conspicuousness of the clues themselves and also what they suggest about the true state of the PCs' reality (or realities).