Abstract: Natural fractures (NFs) are common in shale and tight reservoirs, where staged multi-cluster fracturing of horizontal wells is a prevalent technique for reservoir stimulation. While NFs and stress interference are recognized as significant factors affecting hydraulic fracture (HF) propagation, the combined influence of these factors remains poorly understood. To address this knowledge gap, a novel coupled hydro-mechanical-damage (HMD) model based on the phase field method is developed to investigate the propagation of multi-cluster HFs in fractured reservoirs. The comprehensive energy functional and control functions are established, while incorporating dynamic fluid distribution between multiple perforation clusters and refined changes in rock mechanical parameters during hydraulic fracturing. The HMD coupled multi-cluster HF propagation model investigates various scenarios, including single HF and single NF, reservoir heterogeneity, single HF and NF clusters, and multi-cluster HFs with NF clusters. The results show that the HMD coupling model can accurately capture the impact of approach angle (θ), stress difference and cementation strength on the interaction of HF and NF. The criterion of the open and cross zones is not fixed. The NF angle (α) is not a decisive parameter to discriminate the interaction. According to the relationship between approach angle (θ) and NF angle (α), the contact relationship of HF can be divided into three categories (θ=α, θ<α, and θ>α). The connected NF can increase the complexity of HF by inducing it to form branch fracture, resulting in a fractal dimension of HF as high as 2.1280 at angles of ±45°. Inter-fracture interference from the heel to the toe of HF shows the phenomenon of no, strong and weak interference. Interestingly, under the influence of NFs, distant HFs from the injection can become dominant fractures. However, as α gradually increases, inter-fracture stress interference becomes the primary factor influencing HF propagation, gradually superseding the dominance of NF induced fractures.