The impact of the carrier-envelope phase (CEP) of an intense multi-cycle laser pulse on the radiation of an electron beam during nonlinear Compton scattering is investigated. An interaction regime of the electron beam counterpropagating to the laser pulse is employed, when pronounced high-energy x-ray double peaks emerge at different angles near the backward direction relative to the initial electron motion. This is achieved in the relativistic interaction domain, with the additional requirements that the electron energy is much lower than that necessary for the electron reflection condition at the laser peak, and the stochasticity effects in the photon emission are weak. The asymmetry parameter of the double peaks in the angular radiation distribution is shown to serve as a sensitive and uniform measure for the CEP of the laser pulse. The method demonstrates unprecedented sensitivity to subtle CEP-effects up to 10-cycle laser pulses and can be applied for the characterization of extremely strong laser pulses in present and near future laser facilities.
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