Which is correct: sleep in the nude or sleep nude? I've seen both phrases being used, and Googling this yields no viable results.
Both are correct. There's a subtle difference, especially if you're into psychology.
Sleeping in the nude changes the environment in which one sleeps. Someone could sleep in pyjamas, in the nude, with a hot water bottle, in the afternoon. The preposition distances the person from the act. If someone slept in the nude and referred to it this way, we could imagine that it doesn't happen that often. Environmental behaviors are considered the easiest to change (search for "NLP logical levels" for some models, and bear in mind that they're just models).
Sleeping nude is behavioral, denoting the manner in which someone sleeps. A person might sleep fitfully, nude, deeply, like a baby. It has connotations of a habit, something done regularly. You could imagine that someone who sleeps nude might resist efforts to persuade them otherwise more strongly. Using the word nude as an adjective rather than in an adverbial clause describes a person, with less distance than it does with the preposition.
As a further example, consider someone who says they are a naturist or a nude when they sleep. Using the term as a noun is a statement of identity. This answer I made to how the word "Jew" became pejorative also illustrates the difference.
The choice doesn't exist with the word naked, which is probably why people who use the word nude are choosing to use it instead. It does have more meaning depending on how it's used.
You could try "sleep naked" in lieu of either. Based on the Ngram, "sleep naked" trumps both "sleep nude" and "sleep in the nude."
The same thing happens using "swim" in place of "sleep" (particularly in the past 50 years).
Interestingly enough, with the word "sunbathe," the results are more neck and neck.
I wondered whether this boils down to a difference between "in the nude" and "nude" - irrespective of the verb - or if the verb might actually influence the preferred terminology.
As a side note, I quickly ran out of verbs that could be used with in the nude, but that didn't stop my research. What did I learn? Driving naked seems to be a relatively new phenomenon.
'In the nude' is grammatically incorrect as in is a preposition and you can't be "in the nude" like you can be in a room, jungle or other. I was in the nude is OK because in becomes a phrasal verb like washing up or stripping off. "In a state of nudity", "she was naked", or simply "nude" are better, but anything goes these days.