Can’t remember where I first heard this phrase, but it stayed in my brain as an amusing (if uncomfortable) image.
According to the Cambridge Dictionary, it means to be in a situation where it is difficult to decide what to do, usually because both of your two choices of action would cause problems, e.g.:
UN troops are in a cleft stick: something has to be done for the civilian population, yet to retaliate would surely be the spark to ignite all-out war.
So, two choices, both unappealing.
Like the unusual and the curious?
If you are a lover of unusual words and curious grammar terms, see what a cleft sentence is: https://aceseditors.org/news/2020/grammar-on-the-edge-its-cleft-sentences-that-were-talking-about
Jonathon Owen starts this interesting article with: “Cleft sentences are one of the most common constructions that you’ve probably never heard of, though you almost certainly use them without even thinking about it.”
He goes on to explain how use the cleft sentence to give your writing greater expressive power.
Try it yourself!