I recently read about real ordered fields. Using real closures, I figured out that for each algebraically closed field $C$ of characteristic $0$, there exists a real closed subfield $R\subseteq C$ such that the extension is algebraic (and $[C:R]=2$). This follows very easily using real closures and Zorn's lemma.
Applying this to the field $\overline{\mathbb{Q}}_p$, it follows that there exists an ordered subfield $R\subseteq \mathbb{Q}_p$ such that the extension is algebraic. The field $\mathbb{Q}_p$ cannot be ordered itself (as $-1$ is a sum of squares), it follows that $R\subseteq \mathbb{Q}_p$ is a strict extension. However, $\mathbb{Q}_p$ has no nontrivial endomorphisms, so the extension cannot be normal.
Are there any explicit constructions for $R$? And if the extension is finite, what would be a generator for $\mathbb{Q}_p$?