# Example of a Karp reduction between problems in NP that is not a Levin reduction?

What is an example of a Karp reduction $$f$$ between two problems $$A, B$$ in $$\textbf{NP}$$ such that $$f$$ does not provide a way to transform certificates of one problem into certificates of the other? In other words, such that $$f$$ is not the Karp reduction in any Levin reduction between $$A, B$$?

The accepted answer to this question mentions that it is not always the case that such a Karp reduction yields a corresponding Levin reduction:

"It is true that the well-known Karp reductions between natural problems turn out to be Levin reduction but this does not need to be true in general."

However, I'm struggling to find/devise a concrete ("natural", if possible) example of why this is the case.