Correctly representing convective precipitation remains a long-standing problem in climate models, due to its highly parameterized nature and unclear role of drivers interacting over a wide range of spatial scales. We analyze and compare simulations of Global Storm-Resolving Models, namely the DYAMOND models, using a methodology based on dimensionality reduction and causal inference, to unravel the contribution of large-scale variables and storm-scale dynamics on precipitation distribution. We derive regions of Column Relative Humidity (CRH), which exclude sharp humidity gradients and help define coherent thermodynamic environments, which are subsequently found to control precipitation throughout half of the tropics. The mean CRH is the primary large-scale driver in regions sufficiently large to maintain homogeneity that is unaffected by storms over the 30-day simulation period. The control of mean CRH on precipitation is notably amplified by considering explicitly the intermediate role of the convective area. Moreover, the effect values are consistent across models and quantiles, which could be further employed to constrain GCMs. Our results show that the most extreme intensities (99.9th percentile) cannot be adequately represented without high-resolution data on vertical velocity. However, their effect on precipitation varies considerably across models and precipitation quantiles, making it more difficult to develop a constraint on storm-scale control.