Consider this sentence:
You can manipulate lightning, mist, and wind; traffic with air creatures; and are resistant to electricity damage.
This looks at first glance like a perfectly normal conjunction. It reads a bit clunky, but we have no problem understanding the meaning. We have three verb phrases (whether VP, vP, IP/TP… I don't think it matters here, but I could be wrong) being conjoined with an "and" complementing the auxiliary in the same way a single one of them could be. But the third VP couldn't be a complement of "can". You can come up with similar examples with NP or PP complements, or complements of P instead of V, etc.1
The fact that students are routinely given tests like this one implies that even noticing that something is wrong isn't automatic for native speakers.
So, what's going on here?
It's flatly ungrammatical. People produce these sentences because kids today are stupid and lazy and don't speak proper English.2 And if you think you understand them, well, if you were properly educated, you'd properly fail to understand them. I think this answer is obviously wrong.
It's like syllepsis. According to some definitions, this actually is syllepsis (and/or zeugma),3 but as far as I know, theoretical treatments that explain syllepsis use definitions that don't fit this example.4 But maybe one of those explanations can be broadened to cover all cases of failed parallelism? Something like this: the sentence is ungrammatical, but we don't notice that until we've already parsed and mostly understood the sentence (or produced it aloud or in writing), at which point all that judgment can do is add the feeling of clunkiness. The problem is that this way seems to contradict how grammar is supposed to work in most theories.5 Not noticing that your parse required the same verb to be used in two different senses is one thing; not noticing that you couldn't even parse the sentence is very different.
This is really a nested coordination: a list of two predicates, one of which is itself a list of two things you can do: [You [[can [manipulate…, traffic…]] and [are resistant…]]]. That certainly parses, and makes sense. And you can even explain why it's clunky: there's a null conjunction on the inner coordination (which is legal but often awkward or ambiguous), and the prosody (in speech) or punctuation (in writing) is misleading.6 The problem with this interpretation is that I don't think it's what people are actually thinking when they produce and process these sentences; I think we interpret it as a list of three things.
The list isn't really the complement to "can", it's a level above that. The actual parse is something like [You [[can manipulate…]; [_ traffic…]; and [are resistant…]]], where the "can" fills the gap in the second coordinate, and there is no gap to fill in the third one, and that's fine.7 This works, but feels fishy for some reason I can't put my finger on.
It's ungrammatical, but we routinely repair ungrammatical utterances into grammatical sentences, and don't consciously notice it in many cases. Which is fine, but… what's the grammatical sentence we repair this into? (And, if the answer is the one from (3) or (4), why do we even need to evoke repair here?)
I'm having a hard time coming up with examples that would rule out possibilities 3 and 4, but it seems like they probably exist.
And maybe there's a better explanation than any of these.
1. I opened up a stack of textbooks and RPG manuals to random pages and quickly found multiple examples. I chose this one because it's obviously not a part-of-speech problem: "are resistant to electricity damage" is a VP/IP/TP, just not one that can come after the auxiliary "can", and removing the tense doesn't "correct" it, but instead ruins the meaning. Unfortunately, I stupidly didn't take notes on where the examples came from—but on further searching, I think I've found it. Almost exactly this sentence appears in the Pathfinder core rules PDF (page 41), but with commas instead of semicolons. So I presumably got it from the printed book with the same rules (different revision, so maybe they changed the punctuation at some point to match house style guides). Anyway, the sentence is describing the "granted powers" of clerics who specialize in air, and it clearly means that the cleric is resistant to electricity, not that he might be, or can choose to be, or anything like that.
2. And if it's even more common in professional formal writing than colloquial speech, well, professional writers must be stupid kids too. And if most copy editors don't correct it, they all need to be fired. And if it's been common for centuries, we need to get Doctor Who off the air so kids today can't use time machines to ruin our language retroactively.
3. Most dictionaries say they're synonyms, but most usage guides, etc. disagree—and then give completely contradictory distinctions. "Syllepsis is semantic, zeugma is syntactic" and "syllepsis is grammatical, zeugma is logical" are nearly opposites. And then there's the Oxford guide and their literary dictionary, which say that all parallel coordination is zeugma, and syllepsis is zeugma gone wrong, with a part-of-speech mismatch. The last does seem to be what Johnson meant by "zeugma", but nobody but Oxford seems to use it that way in the 20th century and beyond. Anyway, I don't want to get into that discussion here.
4. In particular, I mean explanations for the pun-like cases where you use a different sense of the same governing verb, or cases like Thurber's "I was losing weight, my grip, and my mind". One explanation is that these problems don't come up during parsing, only during later stages of… whatever happens in semantics, or maybe even after. So we understand the sentence, then notice that the incongruity, which makes the sentence strikes us as funny or powerful (or, when poorly done, clunky). That explanation doesn't immediately seem to work for "You can are resistant to electrical damage", which is just flat-out ungrammatical, not grammatical-but-requiring-a-conflicting-reading.
5. Not just Chomskyan theories. (In fact, they can just say this is a performance rather than competence problem, or that whatever's going on here is part of the periphery, not the core…) It's just as bad for constraint PSGs, CxGs, etc. Any theory that says that we understand sentences by building (or unifying) a parse tree is going to have trouble explaining how we can understand this sentence even though you can't build a valid tree for it.
6. The processing-cost factor of holding both the outer and inner sub-trees in your head until you get to the end of the sentence and can finally disambiguate things could also contribute to clunkiness.
7. Possibly not using a gap when you have something in position to fill the gap is legal but marked, which is part of what makes this clunky?