Precipitation from tropical cyclones (TCs) can cause disasters such as flooding, mudslides, and landslides. Predicting such precipitation in advance is crucial, giving people time to prepare and defend against these precipitation-induced disasters. Developing deep learning (DL) rainfall prediction methods offers a new way to predict potential disasters. However, one problem is that most existing methods suffer from cumulative errors and lack physical consistency. Second, these methods overlook the importance of meteorological factors in TC rainfall and their integration with the numerical weather prediction (NWP) model. Therefore, we propose Tropical Cyclone Precipitation Diffusion (TCP-Diffusion), a multi-modal model for global tropical cyclone precipitation forecasting. It forecasts TC rainfall around the TC center for the next 12 hours at 3 hourly resolution based on past rainfall observations and multi-modal environmental variables. Adjacent residual prediction (ARP) changes the training target from the absolute rainfall value to the rainfall trend and gives our model the ability of rainfall change awareness, reducing cumulative errors and ensuring physical consistency. Considering the influence of TC-related meteorological factors and the useful information from NWP model forecasts, we propose a multi-model framework with specialized encoders to extract richer information from environmental variables and results provided by NWP models. The results of extensive experiments show that our method outperforms other DL methods and the NWP method from the European Centre for Medium-Range Weather Forecasts (ECMWF).