Hull roughness, largely caused by biofouling, alters boundary-layer development and the resulting wake field, thereby affecting the inflow conditions at the propeller plane and the overall propulsive performance. Rather than prescribing roughness and analysing its impact on the wake in a purely forward sense, this work formulates a PDE-constrained inverse identification problem: a spatially varying roughness distribution is identified so that a prescribed, measured or computed target wake field is reproduced in the propeller plane. A steady high-Reynolds-number RANS framework is employed in a fully rough regime, using roughness-based wall functions. The roughness field is determined by minimizing wake-based objective functionals, comprising integral measures of wake uniformity and velocity deficit, as well as a cell-wise wake-matching (band-tracking) formulation for a direct reconstruction of target wake fields. The required sensitivities are obtained using a continuous adjoint formulation, yielding gradients for all roughness parameters at a computational cost comparable to one additional flow solution. Validation is performed through a model-scale wake-reconstruction experiment, achieving a reduction of the initial objective by more than 95\%. Finally, the same adjoint-based inverse roughness identification is performed at full-scale to demonstrate applicability under operational Reynolds numbers. Selected applications include inverse roughness identification in the presence of an actuator-disc model to assess propulsion effects on the wake and propeller inflow.