meaning - Does "I am eating vegan cheese in my underpants" really imply that the vegan cheese is inside my underpants?

I am having a debate with someone about possible interpretations of a sentence and we have come to a stalemate. The sentence is as follows:

"I'm at home eating vegan cheese in my underpants and singing Bushes of Love."

The other party argues that my sentence technically says that I am eating vegan cheese which is inside my underpants, when the intended meaning was that I am eating vegan cheese while wearing underpants.

My argument is that his interpretation of the sentence makes no sense, but I can't quite find an actual rule to refer to. I could see it making more sense if I had an article in front of vegan cheese, like "I'm at home eating the vegan cheese in my underpants and singing Bushes of Love".

But as it is without an article if I did intend to say that the vegan cheese was in my underpants, the way I said it would sound very wrong. It seems to me that without an article in front of vegan cheese it was used as a mass noun, uncountable and therefore interpreting it as being a countable chunk of vegan cheese that could be in anything in any subsequent part of the sentence seems wrong.

His argument is that the problem is in the ordering - that while with some common sense my actual meaning might be clear, the sentence was grammatically incorrect and I technically said that the vegan cheese was in my underpants because I should have said "I am in my underpants eating vegan cheese".

Is there a rule I can refer to when it comes to accurate ways to interpret this sentence? Which one of us is correct? Could the vegan cheese really be in my underpants?

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Research I have done:

My attempts to find a concrete rule that might be relevant to this scenario led me to the following resources. To me none of these resources suggested that my sentence was technically incorrect. However, because I still was not sure what specific rule I could reference for him or if I was missing a specific rule proving me wrong our debate continued. This is why I decided to ask the question here.

Answer

Your friend is wrong: the sentence is grammatically completely correct with the meaning you intended. There is no rule that requires the prepositional phrase "in my underpants" to modify the immediately preceding noun phrase "vegan cheese." Not even a "technical" one. It sounds like the kind of pseudo-rule that would be invented by someone under the misimpression that the rules of English syntax are designed to avoid ambiguity. In fact, this sentence, like many others, is just syntactically ambiguous. That's not a problem; many sentences are. Context makes it clear what you mean, in this case as in many others.

Here is a basic summary from "Linguapress.com Essential English Grammar" of where verb phrase modifiers (like the prepositional phrase "in my underpants") can go in an English sentence:

adverb phrases (groups of words, usually formed starting with a preposition) can come in three possible places:

a) Before the subject (Notably with short common adverbs or adverb phrases, or sentence adverbs - see below) [...]
b1) After the object (virtually any adverb or adverb phrase can be placed here) [...]
c) In the middle of the verb group. (Notably with short common adverbs of time or frequency)

The grammatical ambiguity arises from positioning rules like these and from the fact that prepositional phrases can be used to modify either noun phrases or verb phrases.

There was a recent Language Log post mentioning the issue of "prepositional phrase attachment": Annals of parsing

Two of the hardest problems in English-language parsing are prepositional phrase attachment and scope of conjunction. For PP attachment, the problem is to figure out how a phrase-final prepositional phrase relates to the rest of the sentence — the classic example is "I saw a man in the park with a telescope". For conjunction scope, the problem is to figure out just what phrases an instance of and is being used to combine.

The title of a recent article offers some lovely examples of the problems that these ambiguities can cause: Suresh Naidu and Noam Yuchtman, "Back to the future? Lessons on inequality, labour markets, and conflict from the Gilded Age, for the present", VOX 8/23/2016. The second phrase includes three ambiguous prepositions (on, from, and for) and one conjunction (and), and has more syntactically-valid interpretations than you're likely to be able to imagine unless you're familiar with the problems of automatic parsing.

See also section 1.2 "Ubiquitous Ambiguity" in "Analyzing Sentence Structure," a chapter from Natural Language Processing with Python by Steven Bird, Ewan Klein and Edward Loper.

Syntactic ambiguity is common in all natural languages. It's not feasible to avoid it when constructing a sentence in English, and trying to do so in general will provide no benefits to your writing. Obviously, it's a good idea to avoid ambiguous syntax when there is a real chance of confusion, but that's not the case with your sentence. Your friend obviously knew what you intended: he's deliberately misinterpreting your sentence.

Here's a similar sentence from the Declaration of Independence, which I would say is a document written in a formal style:

In every stage of these Oppressions We have Petitioned for Redress in the most humble terms.

This doesn't mean they wanted redress that was in the most humble terms. It means they petitioned in the most humble terms for redress.

It is neither bad grammar nor bad style to use ambiguous grammatical structures.