Okay, a little more context about phonology. The question appears to be not “can your UR be different from your output?” but rather “can your UR be different from all your outputs?” As in, could you have something like /pana/ -> [pa] and never ever see the /na/ part, in any context?
At first pass, I’m inclined to say no. There’s no reason you would posit something different from what you hear — at least not without overwhelming evidence. Morphophonemic alternations are good evidence, but then usually one of the things appear, just not in all contexts.
This is a debate in modern phonology. Let’s say you’re in a language that uses schwa to break up consonant clusters, but nowhere else. So there will (probably) be words that always always have schwa in them, just exactly where you’d predict based on the phonotactics. Are they there in your underlying representations? It used to be thought that the answer was no; we want to make our URs as simple as we possibly can. Then with OT and the whole Richness of the Base thing, it sort of changed to the answer being yes. You store what you hear, barring information to the contrary. Now I know there is some discussion on how divergent an underlying form can be from the output, but this is usually about one segment each in a few languages, as far as I know. (Look, I’m not a phonologist.)
I can think of a way that you could have a UR that wasn’t identical to any output you have, but shared each part with at least one output. Let’s imagine a language that adds glottal stops to beginnings of words if they start with vowels, and deletes final vowels.
Look at an input /ata/. You need to add a glottal stop, so you get {?ata}. Then you delete the final vowel (order here doesn’t matter) and you get [?at].
Now let’s imagine a prefix, /n-/. Add the prefix, you get /n-ata/. Delete the final vowel, you get [nat].
Why would you supply that final /a/? Well, if you only had prefixes, you wouldn’t. But — hey! — let’s imagine a suffix /-k/. Add that and you get /ata-k/, and finally [?atak]. Assuming you had further data that showed your suffix was /-k/ and not /-ak/ (imaginea form /iti-k/ -> [?itik] etc), you would, once you’re trying to get your copies of each morpheme down to a single form, end up with /ata/, even though you never hear it.
(I lie, a bit. You could hear it in /n-ata-k/, of course. There are more complex examples where you wouldn’t hear it, but this is good enough.)
This is tough work. There is research being done on how learnable it is — but we *do* have morphophonemic alternations, so it must be learnable somehow, even if we don’t know how. A carefully ordered set of rules does not work so well. This is why few people believe in just a carefully ordered set of rules — though as a way to see things, it is not the worst way to begin learning phonology.
Why lie in an intro phonology course? Should we simplify? Yes and no. It’s nice to know the truth, but the reality is sometimes you need a little bit of practice in the basics ofa field before actually learn how things really are. Courses in science are courses in successively smaller lies. Electrons do not circle the atom in nice orbits, no matter how useful a picture that is. But it’s a lot easier to learn that than to learn that a 3p electron has a 95% probability of being in this particular area, at first.
This isn’t to say that linguistics is like chemistry in that; it’s not. Or that linguistics, or MIT and satellite linguistics departments (which do not comprise all the linguistics done in the US, by any means) do not have problems. However, having marginally unfaithful outputs to your input is not one of them. (You are welcome to say that we memorise all the different forms. But you have to say something to explain it.)