Right-angled hexagon tilings of the hyperbolic plane

We study isometry-invariant probability measures on the space $\Omega$ of tilings of the hyperbolic plane with right-angled hexagons of varying shapes. We prove that, for each measure $\mu$ in a certain natural family of measures on right-angled hexagons, there is an isometry-invariant measure on $\Omega$ whose marginal distribution on tiles is $\mu$.