Cross-Sections of Divisible Abelian $o$-Groups via Tame Pairs

It is shown that images of cross-sections of surjective morphisms $f: \Gamma \longrightarrow \Delta$ of divisible abelian $o$-groups are exactly divisible, tame (equivalently, relative Dedekind complete) and cofinal subgroups of $\Gamma$ compatible with $f$ in a suitable sense. The note concludes with an application to real closed valued fields.