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We present a numerical weak-lensing analysis that is fully relativistic and non-perturbative for the scalar part of the gravitational potential and first-order in the vector part, frame dragging. Integrating the photon geodesics backwards from the observer to the emitters, we solve the Sachs optical equations and study in detail the weak-lensing convergence, ellipticity and rotation. For the first time, we apply such an analysis to a high-resolution relativistic N-body simulation, which consistently includes the leading-order corrections due to general relativity on both large and small scales. These are related to the question of gauge choice and to post-Newtonian corrections, respectively. We present the angular power spectra and one-point probability distribution functions for the weak-lensing variables, which we find are broadly in agreement with comparable Newtonian simulations. Our geometric approach, however, is more robust and flexible, and can therefore be applied consistently to non-standard cosmologies and modified theories of gravity.