This paper presents an efficient hybrid continuum-discrete macro-modelling strategy with an enhanced multiscale calibration procedure for realistic simulations of brick/block-masonry bridges. The response of these structures is affected by the intrinsic nonlinearity of the masonry material, which in turn depends upon the mechanical properties of units and mortar joints and the bond characteristics. Finite element approaches based upon homogenised representations are widely employed to assess the nonlinear behaviour up to collapse, as they are generally associated with a limited computational demand. However, such models require an accurate calibration of model material parameters to properly allow for masonry bond. According to the proposed approach, the macroscale material parameters are determined by an advanced multi-objective strategy with genetic algorithms from the results of mesoscale “virtual” tests through the minimisation of appropriate functionals of the scale transition error. The developed continuum-discrete finite element macroscale description and the calibration procedure are applied to simulate the nonlinear behaviour up to collapse of multi-ring arch-bridge specimens focusing on the 2D planar response. The results obtained are compared to those achieved using detailed mesoscale models confirming the effectiveness and accuracy of the proposed approach for realistic nonlinear simulations of masonry arch bridges.