This article proposes a reliability model for systems subject to dependent degradation and external shocks. The dependence between degradation and external shocks is considered as follows. When the system is working but the degradation level reaches a critical threshold, the degradation rate increases in a stochastic sense. Meanwhile, the system becomes more vulnerable to external shocks due to accumulated degradation. An extended degradation-threshold-shock (DTS) model is developed to describe these two competing but dependent failure processes. The continuous degradation path is monotone and modeled by a two-stage inverse Gaussian (IG) process with random effects to account for the unit-specific heterogeneity across systems. External shocks arrive at the system following a doubly stochastic Poisson process (DSPP) with stochastic increasing intensity which depends on the degradation level. Integrating the degradation-based and shock-based failure processes, the system reliability function is derived. Afterwards, a numerical example is presented to illustrate the applicability of the developed reliability model, along with sensitivity analysis to examine the effect of several parameters on the reliability performance. The proposed reliability model is supposed to be applied directly or customized for other complex systems that experience dependent competing failure processes.