Permeability anisotropy of fault rocks has been documented in crystalline and clastic lithologies, but rarely within carbonates. In this contribution, we investigate how a permeability anisotropy may develop within carbonate fault rocks, including deformation bands, for the purpose of improving understanding of fluid flow. A total of 43 oriented fault rock samples plugged in three orthogonal directions were taken from eight faults in differing carbonate lithofacies. The permeability was measured, with the goal of assessing if and to what extent a permeability anisotropy may develop. Key factors controlling the formation of anisotropy in these rocks were analyzed by combining petrophysical and microstructural data. All samples showed some degree of anisotropy. However, a consistent major permeability anisotropy (up to five orders of magnitude) only occurred when the same or similar lithofacies were juxtaposed, where the lowest permeability was recorded normal to fault strike in 75% of the samples. Differences occurred in the highest permeability direction dependent on lithofacies. In deformation bands within high porosity grainstones, the highest permeability was inferred to be at a low angle to sigma 1, created by grain and pore alignment in the direction of transport. The highest permeability in faults cutting recrystallized carbonates varied from sub-parallel to sigma 1, to sub-parallel to sigma 2, owing to variations in Riedel shears and fracture orientation during multiple reactivation episodes. Predicting the permeability of a fault zone, including any directional permeability, is key for improved modeling of fluid flow pathways around faults in the subsurface.