The fully-packed loop model on the honeycomb lattice is a critical model of non-intersecting polygons covering the full lattice, and was introduced by Reshetikhin (1991 J. Phys. A: Math. Gen. 24 2387). Using the two-component Coulomb-gas approach of Kondev et al (1996 J. Phys. A: Math. Gen. 29 6489), we argue that the scaling limit consists of two degrees of freedom: a field governed by the imaginary Liouville action, and a free boson. We introduce a family of three-point correlation functions which probe the imaginary Liouville component, and we use transfer-matrix numerical diagonalisation to compute finite-size estimates. We obtain good agreement with our analytical predictions for the universal amplitudes and spatial dependence of these correlation functions. Finally we conjecture that this relation between non-intersecting loop models and the imaginary Liouville theory is in fact quite generic. We give numerical evidence that this relation indeed holds for various loop models.