High-resolution particle image velocimetry measurements are made in the streamwise–wall-normal plane of a zero-pressure-gradient turbulent boundary layer over smooth and rough walls at Reθ ≈ 13000. The roughness considered herein is replicated from a surface scan of a turbine blade damaged by deposition of foreign materials and its topography is highly irregular and contains a broad range of topographical scales. Two physical scalings of the same roughness topography are considered, yielding two different rough surfaces: RF1 with k = 4.2 mm and RF2 with k = 2.1 mm, where k is the average peak-to-valley roughness height. At Reθ ≈ 13000, these roughness conditions yield k+ ≡ k/y* = 207, δ/k = 28, ks+ = 115, and δ/ks = 48 for RF1 and k+ = 91, δ/k = 50, ks+ = 29, and δ/ks = 162 for RF2 (where δ is the boundary-layer thickness, ks is the equivalent sand-grain height, and y* is the viscous length scale). The mean velocity deficits along with the Reynolds normal and shear stress profiles for both roughness conditions collapse on the smooth-wall baseline in the outer layer when appropriately scaled by the friction velocity, uτ. Probability density functions and quadrant analysis of the instantaneous events contributing to the mean Reynolds shear stress show similar outer-layer consistency between the smooth and rough cases when scaled appropriately with uτ. In addition, one-dimensional, two-point streamwise, and wall-normal velocity autocorrelation coefficients are also found to collapse in the outer region, indicating a similarity in the spatial structure of the outer-layer turbulence. The observed collapse of the smooth- and rough-wall turbulence statistics in the outer layer supports Townsend’s wall similarity hypothesis for flow over the unique surface topography considered herein.