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Permeability anisotropy of fault rocks has been documented in crystalline and clastic lithologies, but rarely within carbonates. Here, conditions for a permeability anisotropy to develop within carbonate fault rocks are documented. Forty-three fault rock samples plugged in three orthogonal directions were taken from eight faults in differing carbonate lithofacies. The permeability was measured, to assess if and to what extent a permeability anisotropy may develop. All samples showed some degree of anisotropy. However, a systematic major permeability anisotropy (up to 5 orders of magnitude) only occurred when the same or similar lithofacies were juxtaposed, where the lowest permeability was recorded normal to fault strike. Differences occurred in the highest permeability direction dependent on lithofacies. In deformation bands cutting high porosity grainstones, the highest permeability was inferred to be at a low angle to σ1, created by grain and pore alignment in the direction of transport. The highest permeability in faults cutting recrystallised carbonates varied from sub-parallel to σ1, to sub-parallel to σ2, 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 modelling of fluid flow pathways around faults in the subsurface.