This is not an answer to my question but it is just a summary of the relevant results which have been mentioned so far about positive solvability.
So here is a useful result:
Theorem Let $K/F$ be a finite Galois extension and let $M/F$ be a radical extension. Assume that you have an embedding $\iota:K\hookrightarrow M$. Then if $[K:F]$ is odd there exists a root of unity of odd order inside $M$.
A proof of this result may be found for example in Brian Conrad's note: radical tower and roots of unity.
We thus have the following corollary
Corollary Let $f(x)\in\mathbf{Q}[x]$ be an irreducible polynomial of odd degree which has all its roots in $\mathbf{R}$. Let $K$ be the (abstract) splitting field of $f(x)$. Then $K$ cannot be embedded in a radical extension contained in $\mathbf{R}$. In particular, $f(x)$ is not positive solvable.