We use three-dimensional X-ray imaging to investigate steady-state three-phase flow in a mixed-wet reservoir rock, while measuring both relative permeability and capillary pressure. Oil occupied the smallest pores, gas the biggest, while water occupied medium-sized pores. We report a distinct flow pattern, where gas flows in the form of disconnected ganglia by periodically opening critical flow pathways. Despite having capillary-controlled displacements, a significant fraction of the pore space was intermittently occupied by gas-oil and oil-water phases. Both types of intermittency occurred in intermediate-sized pores. Gas mainly displaces oil, and oil displaces water as the gas flow rate is increased, while oil displaces gas, and water displaces oil as gas flow is decreased. At the resolution of the images, no detectable gas was trapped in the rock due to its mixed-wettability which prevents either oil or water completely surrounding gas, suppressing snap-off and capillary trapping, which has significant implications for the design of gas storage in three-phase systems.