Fish fish fish eat eat eat <- for real?

This is what one paper calls a "multiple self embedded sentence":

We have [a self-embedded (SE)] sentence when a phrase is placed totally within another phrase of a similar type, e.g., (1) The nurse that the cook saw heard the butler. A multiple SE sentence has a phrase within a phrase which is in another phrase, e.g., (2) The nurse that the cook that the maid met saw heard the butler. The latter example is definitely unacceptable to speakers of English. People have difficulty in pairing the verbs with their subject nouns, if indeed they even get as far as recognizing that the words comprise a sentence.
_{Observations with self-embedded sentences}

(The other thing that is happening in your sentence is that "that" is being dropped.)

These are theoretically valid sentences, but are very hard to parse and therefore are unacceptable to most people.

You can create similar sentences yourself by writing write a valid sentence using a placeholder pronoun ("the man {she} loves is gone"), then replace that pronoun with something. Said something that's replacing the pronoun can be another phrase, itself using a placeholder pronoun ("a woman {he} knows"), and then you can repeat this infinitely ("the man {a woman {a child {another boy} saw} knows} loves is gone").

I can make this work grammatically, but you REALLY have to squint at it.

I'll break it down for you.

Fish eat. This is a simple sentence. Fish (n) eat (v).

Fish fish eat This is not a sentence. It's a noun phrase. (Fish that other fish eat.)

Fish fish fish eat. This is a sentence. [Fish that other fish fish for] (noun phrase) eat (v).

Fish fish eat eat. This is a sentence. Fish who are eaten by other fish (noun phrase) eat (v).

Fish fish fish eat eat. The fish which (fish who are eaten by other fish) eat. -- Noun phrase.

Fish fish fish eat eat eat. - The fish which get eaten by (fish who are eaten by other fish) (very complex noun phrase) eat(v).

This is comparable with the famous buffalo sentence.

I won't unravel that here. But, it's worth a read.