As discussed in this question, in general a Borel surjection $f:\mathbb{R}\rightarrow\mathbb{R}$ may not have a Borel right inverse, namely a $g$ such that $f\circ g=id$, although there is always a coanalytic right inverse by Kondo uniformization theorem.
Meanwhile, near the end of Becker's article Descriptive set theoretic phenomena in analysis and topology, the author says he knows of no natural such example in analysis or topology (the article gives many interesting examples of other phenomena). In fact he makes a stronger claim: there is no natural example of a Borel $R\subseteq\mathbb{R}\times\mathbb{R}$ that satisfies $\forall x\exists y\ R(x,y)$ and has no Borel uniformization. He also says this is related to reverse mathematics, and is "more or less equivalent" to finding a statement $\forall x\exists y\ R(x,y)$ about reals that is true but not provable in Kripke-Platek set theory.
Questions:
- Do we have any natural example now?
- Can someone explain the relation to reverse mathematics and $KP$?
