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We examine the evolution of ion-beam Weibel instability at strong collisionless shocks in weakly magnetized media. We find that a finite background magnetic field substantially affects both the linear and nonlinear phases of the instability, depending on whether the background electrons behave magnetized or not. Particle-in-cell simulations for magnetized electrons identify a dynamo-like mechanism of magnetic field amplification, which eventually leads to spontaneous magnetic reconnection. We conclude that this scenario is applicable to typical young supernova remnant shocks.