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Inferring causal relations from observational time series data is a key problem across science and engineering whenever experimental interventions are infeasible or unethical. Increasing data availability over the past decades has spurred the development of a plethora of causal discovery methods, each addressing particular challenges of this difficult task. In this paper we focus on an important challenge that is at the core of time series causal discovery: regime-dependent causal relations. Often dynamical systems feature transitions depending on some, often persistent, unobserved background regime, and different regimes may exhibit different causal relations. Here, we assume a persistent and discrete regime variable leading to a finite number of regimes within which we may assume stationary causal relations. To detect regime-dependent causal relations, we combine the conditional independence-based PCMCI method with a regime learning optimisation approach. PCMCI allows for linear and nonlinear, high-dimensional time series causal discovery. Our method, Regime-PCMCI, is evaluated on a number of numerical experiments demonstrating that it can distinguish regimes with different causal directions, time lags, effects and sign of causal links, as well as changes in the variables' autocorrelation. Further, Regime-PCMCI is employed to observations of El Niño Southern Oscillation and Indian rainfall, demonstrating skill also in real-world datasets.