What is it called when you combine two distinct phrases with a common middle phrase or word?

Question

Is there a term for when you combine two distinct phrases together with a middle term?

For example:

• Big Band Aid (Big Band/Band Aid. Alternately, a benefit concert for swing musicians)

• Insane Asylum Seekers (Crazy, but at least they're free)

I'm thinking it might be an apo koinou construction, but the examples of that I've found are more complex, and I'm not sure if they're really the same thing.

I found this thread, but it doesn't quite answer the question:
A figure of speech combining two phrases

Answer 1

I would call these phrases blends.

According to Merriam-Webster's Dictionary (https://www.merriam-webster.com/dictionary/blend):

BLEND

: a word (such as brunch) produced by combining other words or parts of words.

Your case is a peculiar one.

Up to now I've met only examples of the so-called 'lexical' blends.

But in the question there are two examples of the so-called 'phrasal' blends.

It's a very interesting model of the phrase formation which is very similar to blending in the vocabulary (like smog, motel, Spanglish, etc.).

I think, to term this figure '(PHRASAL) BLENDING' would be appropriate.

As about apo koinou, it is also a blend, but that of two sentences (predicative constructions), and is defined as:

''the occurrence of one and the same word or word group, not repeated, in two constructions

(such as three crows in “there were three crows sat on a tree”)'' (https://www.merriam-webster.com/dictionary/apo%20koinou).

In the question there are no sentences, but nominative phrases.

Apo koinou is a syntactic stylistic figure when they omit a conjunction or relative pronoun necessary for the standard complex sentence ('which' in the sentence from the Dictionary entry).

That's why I would use the term 'blending'.

Answer 2

Perhaps the term you are looking for is Malaphor. It's similar to what you're talking about. Malaphor is a portmanteau of malapropism and metaphor. It typically applies to combining metaphors like "Let's burn that bridge when we come to it" or "Don't count all of your chickens in one basket." https://www.thoughtco.com/malaphor-word-play-1691298

Answer 3

These are commonly referred to in U.S. Television game shows as "Before and After"

https://wheeloffortuneanswer.com/before-and-after/
https://jeopardyquestions.com/category/before-after