Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is *$F$-archimedean* if for each $e\in E$ there is $x\in F$ such that $-x\le e\le x$.

Is it true that if $E$ is $F$-archimedean then every interval in $E$ contains an element in $F$? That is, is it true that for every $e<e'$ in $E$ there is an element $x\in F$ such that $e<x<e'$?

This is known if $F$ is the field of real algebraic numbers (in which case $E$ is an ordered subfield of $\mathbb{R}$), and it seems to me that it should have an easy proof in the general case. However I cannot find neither an easy proof nor a counterexample.