Magnetic reconnection associated to the double tearing mode (DTM) is investigated by means of resistive relativistic magnetohydrodynamic (RRMHD) simulations. A linearly unstable double current sheet system in two dimensional cartesian geometry is considered. For initial perturbations of large enough longitudinal wavelengths, a fast reconnection event is triggered by a secondary instability that is structurally driven by the nonlinear evolution of the magnetic islands. The latter reconnection phase and time scale appear to weakly depend on the plasma resistivity and magnetization parameter. We discuss the possible role of such explosive reconnection dynamics to explain the MeV flares observed in the Crab pulsar nebula. Indeed the time scale and the critical minimum wavelength give constraints on the Lorentz factor of the striped wind and on the location of the emission region respectively.