Wess-Zumino-Witten (WZW) models are among the most basic and most studied Conformal Field Theories (CFT). They have had a huge influence not only in physics but also in mathematics, in representation theory and geometry. However their rigorous probabilistic construction and analysis starting from the path integral is still missing and all their properties have been obtained algebraically from their postulated affine Lie algebra symmetry. Initially considered as taking values in a compact semisimple Lie Group G, the WZW model also has a "dual" formulation where the group $G$ is replaced by the homogenous space $G^{\mathbb{C}}/G$, where $G^{\mathbb{C}}$ is the complexification of $G$, and it has been argued that the former can be (re-)constructed from the latter. For $G={\rm SU}(2)$, the space ${\rm SL}(2,\mathbb{C})/{\rm SU}(2)$ can be identified with the three dimensional hyperbolic space $\mathbb{H}^3$ and, in physics, the corresponding CFT has been studied as the simplest example of the AdS/CFT correspondence. A surprising correspondence between the $\mathbb{H}^3$-WZW CFT and the Liouville CFT was found by Ribault and Teschner and later generalised by Hikida and Shomerus. This correspondence has been dubbed by Gaiotto-Teschner as the "quantum analytic Langlands correspondence" since the analytic Langlands correspondence of Etingof, Frenkel and Kazhdan seems to emerge in its formal semi classical limit. In this paper we give a rigorous construction of the path integral for the $\mathbb{H}^3$-WZW model on a closed Riemann surface $Σ$, twisted by an arbitrary smooth gauge field on $Σ$. Using the probabilistic path integral we prove a correspondence between the correlation functions of the primary fields of the $\mathbb{H}^3$ model and those of Liouville CFT extending the expressions proposed by Ribault-Teschner and by Hikida-Schomerus to this general setup.