Abstract Ternary mixtures of saturated and unsaturated lipids together with cholesterol can be induced to phase separate by photo-peroxidation into lipid-ordered Lo and lipid-disordered Ld domains. Because these have different mechanical properties, the phase separation is accompanied by dramatic changes in morphology. This work considers a tubule composed of Ld phase with Lo phase inclusions that possess greater rigidity; this system has been shown experimentally by Yuan and coworkers to spontaneously adopt either banded or disc configurations following phase separation. The static behaviour of inter-domain interactions is analyzed in each of these geometries by solving the linearized shape equations. These calculations suggest a possible mechanism by which the two structures form.