Let $\mathcal{M}=(M,<,\ldots)$ be an o-minimal structure, namely a linearly ordered (by $<$) first order structure such that every definable set in $M$ is finite union of points and intervals $(a,b)$ where $a\in M\cup\{-\infty\}$ and $b=M\cup \{+\infty\}$.
Is there an expansion of $\mathcal{M}$ to an o-minimal structure that has definable choice (i.e. definable Skolem functions)?
Extra questions:
What about an expansion to an o-minimal structure that expands an ordered group $(M,0,+,<)$, or an ordered field $(M,0,1,+,\cdot, <)$?