This is an object that, in order to see it (or hear it, touch it, etc.), you have to already believe it's there. However, this doesn't change the fact that they are there, even if you have no way to sense them.
Let's say The Hero is convinced that the magical MacGuffin which can defeat the Big Bad doesn't exist, or that it was some kind of lame metaphor. But, at the eleventh hour, the Big Bad still recoils in horror at the sight of the Mac Guffin's treasure box. It turns out that the "empty" box The Hero has been carrying around all this time actually does have the MacGuffin! Only, The Hero still can't see it, touch it, or even interact with it. It's there for everyone else, but not for him.
Another example is a person or entity that everyone's been trying to get another character to believe in, only for them to dismiss it. When the door slammed on its own, it was "just the wind".
This trope is in a similar vein to Gods Need Prayer Badly and Clap Your Hands If You Believe. Like in those tropes, belief in the metaphysical is key. However, unlike those tropes, the thing that's being imagined was there all along, but in a form that is useless to human senses. It isn't until you believe it's there that it finally makes itself manifest to you.
Compare Enlightenment Superpower, Power Born of Madness, Achievements in Ignorance, Imaginary Friend, Not-So-Imaginary Friend, Invisible to Normals and Living Dream.
NOTE: Again, what makes this different from Clap Your Hands If You Believe is that believing in something isn't what causes it to exist. It existed all along and (usually) would continue to exist even if the other characters stopped believing. It would simply remain Invisible until belief returned.
Fairies and elves are this way in Berserk. There's clearly a scene in which Puck is interacting with confused non-believers, who can't see him even though he's right before their eyes. Those that believe in fairies, however, can see him with no problem.
In Higurashi, there is a deity called Oyashiro-sama that people believe in. They think a lot of things about it that simply aren't true. For instance, they call Oyashiro a guy, only the REAL oyashiro is....a FEMALE. Not only that, some people, like Takano, believe Oyashiro-sama approved of the horrible deeds that the villagers committed in the past. In reality, Takano is the true monster, while Oyashiro-sama is nothing more than an innocent girl goddess who wishes for peace for her friends. However, the for the majority of the series, Hanyuu is there, but invisible to everyone except for Rika and people who are at a high enough level on the Hinamizawa Syndrome.
The banquet in Hook. Everyone starts gorging themselves around Peter, and he can't understand it. When they tell him to simply imagine the food, it finally appears. No, actually, judging by everyone's surprise after it happens, his imagination overrides theirs.
The title character in Drop Dead Fred is an Imaginary Friend who was sealed in a Jack-In-The-Box for many years, until his imaginator(?) accidentally releaeses him. Presumably if he hadn't been sealed up he'd've been out in the world even if he had no longer been believed in.
In The Santa Clause, in the Workshop, Judy the elf states this to Scott, adding, "kids don't need to see this place to know that it's there. They just... know."
In the Nellie and the Dragon books by Elizabeth Lindsay, nobody can see Gertrude the Dragon unless they already believe she exists. At first, that's just Nellie, but later her best friend also comes to believe. Gertrude can still interact invisibly with people who don't believe in her, which gets Nellie into quite a bit of trouble.
The short story "Faith" by Sherwood Smith revolves around this.
The children's book O'Diddy by Jocelyn Stevenson is built on this torpe, combined with a dose of Accidental Nightmare Fuel: when an imaginary friend's child partner grows too old to believe in them, the imaginary friend is still around, but invisible and unable to interact with them in any way. The only way to avoid an And I Must Scream situation is to seek out another child to befriend. Oh to do what O'Diddy does at the end, and convince the grown up child to believe again.
Unicorns in The Last Unicorn are visible as themselves only to those who believe they exist — otherwise they look like horses.
Prince Caspian: When the Pevensie children are returned to Narnia for the first time, at first only Lucy sees Aslan. Edmund doesn't see Aslan but he believes her (since she was right about Narnia the first time) but the older Peter & Susan veto Lucy, thinking that either she's fibbing or imagining things. But after they've been lost for hours they finally give in and let Lucy lead the way - Lucy is following Aslan. Slowly, Edmund & Peter start to see him. Eventually even Susan sees him, even though she sees him before she admits to it.
In Doctor Who, Perception Filters are explained to work to this way. If you don't want to/don't have a reason to believe there's a blue anachronistic police box on the street corner, a second floor on your house that shouldn't be there, etc., then you simply won't see anything. But if you do want to believe there's something there, the perception filter falters and you become aware of the thing it was shielding.
Japanese Mythology has this system in place for the "kami", the gods/spirits/ancestors that control natural phenomena. In Shinto mythology, the kami are Anthropomorphic Personifications of natural phenomena, objects, or concepts that reside in an "invisible world" outside of human perception, but that is still part of the real world. In order to see them, the first step is to already believe they're there, but that's not the only requirement. Only the Pure of Heart will fully be to see and interact with them.
In Ōkami: The protagonist, Amaterasu, can be seen by the other characters but her facial markings can only be seen by those who still believe in the gods. Those who can't assume she's just an ordinary wolf. There are two sisters, the younger of whom believes in the gods (and can see Ammy's true form) while her older sister cannot. If you perform enough Celestial Brush powers for her, though, the older sister does start to believe, and will comment that she can see "faint" markings on Ammy.
In Umineko no Naku Koro ni, witches like Beatrice, and magic in general, can supposedly only be seen by those who genuinely believe in them in the human world. That's because they don't actually exist in the human world.
Played with in the webcomic Count Your Sheep, where Katie's "imaginary friend" Ship is real, but can only be seen by certain people. At first that's Katie and Katie's mom Laurie (whose imaginary friend Ship was when she was Katie's age), but later the strip introduces Laurie's mom — who can see Ship, but repeatedly denies that she believes he exists even when she's talking to him and there's nobody else around — and Laurie's Cool Big Sis — who can't see Ship, even though she believes he exists.
In one episode of Teen Titans Raven guards three super powered children all the while telling the oldest one her imaginary friend isn't real but at the end of the episode he's what saved all their lives.
Five hats means that five tropers think it is ready to publish.
You are saying that you think this draft is ready to be published. That means the description is not ambiguous,
it doesn't duplicate an existing trope, there are at least three examples, and the title makes sense.
Is that what you meant to do?
You are saying this draft has a ready-to-publish hat it does not deserve and you are taking it back.