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Co/Pt multilayers can exhibit considerable exchange bias along the film normal when grown onto or covered by an antiferromagnetic IrMn layer. The magnitude of the bias effect depends strongly on the selection of the buffer layers and is largest for Co/Pt multilayers with small perpendicular anisotropy. This counterintuitive result is explained by opposite dependencies of exchange bias and magnetic anisotropy on the degree of film texture and grain size. Interface roughness also influences the perpendicular exchange bias field. The largest biases are measured on smooth films. Interlayer mixing during post-deposition annealing procedures decreases the exchange bias field of IrMn-capped Co/Pt multilayers.