Is the image of Borel measurable function essentially Borel measurable?

Question

It is known that whenever $f: \mathbb{R} \rightarrow \mathbb{R}$ is Borel measurable it is not necessarily true that $f(X)$ is Borel measurable for Borel $X \subset \mathbb{R}$. However I have not seen a counterexample yet.
It is known that whenever $f$ is injective the claim holds.
Question: Does there exists a Borel function $\tilde{f}: \mathbb{R} \rightarrow \mathbb{R}$ such that $f = \tilde{f}$ a.e. and such that $\tilde{f}(X)$ is Borel measurable for each Borel $X \subset \mathbb{R}$?