  Motivated by a recent adsorption experiment [M.O. Blunt et al., Science 322,
1077 (2008)], we study tilings of the plane with three different types of
rhombi. An interaction disfavors pairs of adjacent rhombi of the same type.
This is shown to be a special case of a model of fully-packed loops with
interactions between monomers at distance two along a loop. We solve the latter
model using Coulomb gas techniques and show that its critical exponents vary
continuously with the interaction strenght. At low temperature it undergoes a
Kosterlitz-Thouless transition to an ordered phase, which is predicted from
numerics to occur at a temperature T \sim 110K in the experiments.
