  We consider close-packed tiling models of geometric objects -- a mixture of
hardcore dimers and plaquettes -- as a generalisation of the familiar dimer
models. Specifically, on an anisotropic cubic lattice, we demand that each site
be covered by either a dimer on a z-link or a plaquette in the x-y plane. The
space of such fully packed tilings has an extensive degeneracy. This maps onto
a fracton-type `higher-rank electrostatics', which can exhibit a
plaquette-dimer liquid and an ordered phase. We analyse this theory in detail,
using height representations and T-duality to demonstrate that the concomitant
phase transition occurs due to the proliferation of dipoles formed by defect
pairs. The resultant critical theory can be considered as a fracton version of
the Kosterlitz-Thouless transition. A significant new element is its UV-IR
mixing, where the low energy behavior of the liquid phase and the transition
out of it is dominated by local (short-wavelength) fluctuations, rendering the
critical phenomenon beyond the renormalization group paradigm.
