Quasi-Fuchsian groups and complex realisations of $q$-deformed real numbers

We relate the theory of $q$-rational and $q$-real numbers introduced by Morier-Genoud and Ovsienko to the classical theory of Kleinian groups and their Teichmüller spaces. This provides a geometric point of view on several recent results about realisations of $q$-rationals for particular values of $q\in \mathbb{C}$. As an application we resolve a conjecture of Bapat, Becker, and Licata on the topology of the $q$-deformed Farey tessellation.