A few years ago, I happened to come across a commonplace "theory" centering around the issue of whether or not Koishi dreams. It has inexplicably become popular fan consensus that Koishi is incapable of dreaming--despite the fact that we have direct evidence that she does. Regardless, I wanted to write a short post disproving this idea since I feel it reflects some reaccurent issues pertaining to how the fanbase at large treats Koishi! I don't think there's any other character within the franchise that has such silly conceptions about her. To begin with, we have to examine how dreaming works within the universe of the series.
Within the Touhou universe, dreaming isn't as simple as just falling asleep. Characters who occupy the universe have alternative versions of themselves, referred to as "Dream Selves" that are immaterial, non-physical beings that exist in what is called the "Dream World." Whenever the regular characters dream, their Dream Selves will access the Dream World, living out their fantasies as they please. The plural version of many Dream Selves are called "Dream Dwellers," and it is also possible for a person to enter the Dream World while awake and with their physical body--it is just very uncommon. (1) (2)
One character is the manager of the Dream World, Doremy Sweet. Doremy is actually familiar with several characters, even those she has never met personally before, because she has encountered their Dream Selves within her domain. A large portion of Doremy's dialogue consists of her discussing the dreams of other characters, as she is able to recall with notable accuracy what in particular it is that they dream about; and so it might be a surprise when Doremy says to Koishi in the game Antinomy of Common Flowers (AoCF) this line: "Hm? You don't dream at all? Your heart is so hollow..." (3)
At first, this seems pretty irrefutable--Doremy has no motive to lie here, and she has plenty of authority to make such a claim with confidence. The real issue is that the Dream Self version of Koishi, Dream Koishi, does exist. She appears in the game Violet Detector on Nightmare Wednesday and is referred to as a dream dweller like the rest of the cast; in other words, there is no way that this is regular Koishi who traveled to the Dream World by physical means. This is her Dream Self. (4)
Everyone in the Touhou fanbase is currently unable to reconcile this very simple logic, and they invent bizarre and easily refuted speculation about Koishi in an effort to justify their incorrect assumption. This is actually something I've noticed quite a lot; the fanbase will start out with their hypothesis as a unequivocal fact--that Koishi doesn't dream, or doesn't have a consciousness--and then when they encounter evidence that directly disproves this, they will try to write fanfiction instead of recognizing that the truth may be more complicated than once thought. One example I heard once is the idea that, maybe, this "Dream Koishi" is actually Koishi before she closed her third eye! Oh, it's a poor lost little version of Koishi's consciousness, isolated from her own self...The issue of course being the fact that Dream Koishi's third eye is still closed in game. I guess you just have to ignore that part.
This leaves the question: why would Doremy come to this conclusion if it isn't true? The existence of Dream Koishi logically necessitates that she does dream--in the almost 10 years since AoCF has been released, we have not been given a single shred of evidence that suggests you can have a "Dream Self" without dreaming. Until it's revealed to us that it is possible to have a Dream Self without dreaming, this is just short of outright confirmation that Koishi does dream. Something has to be amiss here, and this definitely isn't the first time ZUN has written Koishi in such a contradictory manner. Several times throughout the series ZUN has written many other characters, such as Akyuu and Kanako, to purposefully come to wrong conclusions about Koishi only to later have their thinking proven wrong by the omniscient narrator. I'd like to offer a few theories as to why this might be the case with Doremy:
The answer may lie in a small line from one of Koishi's prior official profiles: the one that asserts that she lives without any dreams, or hopes. This line specifically refers to "having dreams" instead of the actual verb of dreaming as Doremy speaks of, but that takes us back to the specific sort of dreams Dream Dwellers have: aspirational ones. Across almost all Dream Dwellers, their dreams focus on strong motives, hopes, and wishes the characters themselves dream of--for example, Reimu's Dream Self is always relaxing, Ichirin's is very respected, and Shinmyoumaru's is gigantic, much larger in comparison to her real-world stature. If Doremy is looking for one of these aspirational dreams in Koishi, she'd find none! This also explains her following comment on her heart being "hollow." I believe this is the most likely answer, but still wanted to discuss a few more!
We still don't know how exactly Doremy assesses the dreams of other characters, as it has never been stated directly to the audience--if Doremy does this through a psychic ability, then it wouldn't work on Koishi because her heart and mind are sealed away. Other characters in the series, such as Satori and Reisen, are incapable of applying their ability to Koishi for this reason as well. Thus, it is possible that some quality of Koishi is preventing Doremy from an accurate assessment.
It is confirmed that it is specifically when Koishi enters an unconscious state does it become difficult for others to recognize, identity, and remember her. If Dream Koishi is Koishi in a purely unconscious state, as people usually are when they are asleep or dreaming, then Dream Koishi might be completely undetectable to Doremy--the two might have never even encountered another. It's also possible they have met, but Doremy has forgotten due to Koishi's influence. There are some problems with this theory--mainly being the fact that Dream Koishi is clearly recognized to be Koishi by Dream Sumireko--but it could be possible.
This is a plot hole or retcon? Actually, I have no idea what's going on. I thought Koishi didn't dream for awhile because of this line and was always confused by it, so this was a surprise for me. Violet Detector comes after AoCF, so if this is a retcon, it's in favor of Koishi dreaming, but I don't think ZUN retcons things often. Anyways, I think Koishi dreams about dinosaurs, microbes, and fish. Good luck figuring this out!!
My favorite artist, Menotama, once wrote a phenomenal short comic about Koishi having a deep, meaningful dream--yet one that she forgets just as she opens her eyes and wakes up, the meaning of the dream itself vanishing with it. I loved the comment they made at the very end, speculating as to Doremy's line: I'm paraphrasing here, but their reaction was to just shrug what she said off and assume she didn't know what she was talking about! After all, why wouldn't Koishi dream? In the western Touhou fanbase, where Koishi is some braindead yandere psychopath, it seems common sense that she wouldn't since she also lacks other mental facets. But in truth, we actually don't have any reason to suppose that Koishi wouldn't be capable of it, besides this singular line of dialogue.

How are theories even supposed to work to begin with? You make a hypothesis out of actual, canon information. It begins with you making an observation about the work itself, investigating it, and then reassessing how accurate you were. But this isn't true anymore, not with Koishi, the Touhou fandom, or most fandom at large. What people do now is start out with an opinion they'd like to be true--or one that they arbitrarily assume to be true--and then work backwards to justify it in spite of lacking evidence or outright refutation. If you were to ask them why they'd come to their warped conclusion, the answer is: Nothing. These people do not consume or care about canon media--the intended meaning and story the author wants you to appreciate. They just want their "cool idea" to be true.
As for me, I've always had mixed feelings about ZUN's decision here! I actually really enjoy that he writes Koishi in a contradictory way, but my main issue has always been...isn't he aware of the connection between dreams and the subconscious? Well, yes! (6) ZUN actually has commented on Jung's idea of the collective unconscious and it's relation to our dreams. So then, why would he miss such an obvious chance to have the Subconscious Character and the Dream Character to interact? Where's Koishi and Doremy's Duo-Spellcard of the name "The Ineffable Subject of the Unconscious"? I know I'm repeating myself here, but I can't be the only person who was a little disappointed in this.

And more importantly, fans have always been speculating not on whether or not Koishi does dream--but what she dreams of, as has been expressing quite beautifully in songs and art. Some of the best fanmade content we've gotten of Koishi has involved her finding herself within a dream, one that mirrors her inner conflicts; whether it's the closure of her own heart, the loss of her emotions, or her fears surrounding intimacy and dislike. All of these facets of Koishi are things she actively pushes out of her conscious, aware mind, things she tries her hardest to ignore--and so it's only natural they'd show up in the realm of the subconscious. And because it's a dream, as an artist, you've been given pretty much no limitations! You have total control over the story you want to tell, but also what sort of abstract imagery you want to express it through. In other words, the idea of Koishi confronting uncomfortable truths about herself within her dreams has always had great potential...Why throw it all away? Why make it impossible?
I'm of the same line of thinking as Menotama, where I believe that Koishi dreams, but certainly not of the same things others might see. In fact, since we're speculating about Dream Koishi a little bit, I actually don't think there's too much of a difference between Real Koishi and Dream Koishi; this is true in an outward sense, as Koishi has no aspirations or strong desires that Dream Dwellers are usually passionately obsessed about. In terms of her daily life as well, Koishi's workings are comparable to that of a lucid dream--since nobody can notice her, she's free to act on any one of her fleeting whims and desires without reprisal. It's like when you realize you're dreaming, so you can do whatever you want! Nobody will stop you, and it isn't real anyways, so just go wild.
But I also imagine it applies in an inward sense--that Koishi likely feels as if there's little difference between a dream or waking reality, since she has lost touch with herself and the world outside of her since closing her third eye. Menotama actually writes Koishi to say a rather striking line towards the end of the comic: that everything feels the same now, no matter if she's dreaming or awake.