  We discuss a Coulomb gas realization of $n$-point correlation functions in
the $SL(2,\mathbb{R})$ Wess-Zumino-Witten (WZW) model that is suitable to
compute scattering amplitudes of winding strings in 3-dimensional Anti-de
Sitter space at tree-level and one-loop. This is a refined version of
previously proposed free-field realizations that, among other features,
accomplishes to make the $H_3^+$ WZW-Liouville correspondence manifest.
