On the Riemann-Hilbert problem for a $q$-difference Painlev\'e equation

A Riemann-Hilbert problem for a $q$-difference Painlev'e equation, known as $q{\text{P}}_{\text{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q{\text{P}}_{\text{IV}}$ and corresponding data on an associated $q$-monodromy surface. We also construct the moduli space of $q{\text{P}}_{\text{IV}}$ explicitly.