As the impact of the correlation and uncertainty of photovoltaic (PV) and wind power generation on electricity- gas-heat regional integrated energy systems (RIES) becomes more pronounced, analyzing the probabilistic energy flow of the electricity-gas-heat RIES can provide an intuitive reflection of its operational state changes. First, considering the correlation between PV and wind power generation and the uncertainty of random variables, an energy flow calculation model for an electricity-gas-heat RIES that includes PV and wind power is established. Then, an stochastic response surface method (SRSM) optimized by the Galerkin projection method is proposed to calculate the RIES energy flow, and the probability distributions of the state variables of the energy flow are obtained to quantitatively assess the impact of the correlation between PV and wind power on the probabilistic energy flow of the RIES. Finally, the proposed model and the solution method are validated using an RIES formed by the IEEE 30-node power system/IEEE 69-node power system, 20-node heat system, and 24-node natural gas system. The validity and applicability of the proposed model and the accuracy and speed of the solution method are verified by analyzing the effects of the correlation between PV power and wind power on the probabilistic energy flow.