This paper proposes an adaptive subregion-based active Kriging (AS-AK) surrogate modeling approach. Firstly, an adaptive subregion decomposition strategy is developed to partition the candidate sample space into multiple concentric subregions, significantly enhancing the efficiency and accuracy of sampling. Subsequently, an active Kriging surrogate model is constructed, where the surrogate model is sequentially updated by iteratively selecting critical samples within each subregion to precisely approximate the highly nonlinear limit state function. Moreover, a collaborative multi-output surrogate modeling framework is further established to systematically handle correlations among multiple failure modes. Four benchmark numerical examples and an engineering application involving an aeroengine rigid-flexible coupling system illustrate that the proposed AS-AK method significantly outperforms existing reliability methods in both computational efficiency and accuracy.