The double dimer model is a variation of the classical dimer model consisting in superimposing two independent dimer configurations (perfect matchings) on a graph, thus creating an ensemble of non-intersecting loops. Kenyon has recently introduced and studied "anyonic" correlators for this model. We discuss the convergence (in the small mesh limit) of some of these correlators to the tau-functions appearing in the theory of isomonodromic (SU(2)) deformations.