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Random environmental variation, or stochasticity, is a key determinant of ecological dynamics. While we have some appreciation of how environmental stochasticity can moderate the variability and persistence of communities, we know little about its implications for the nature and predictability of ecological responses to large perturbations. Here, we show that shifts in the temporal autocorrelation (colour) of environmental noise provoke trade-offs in ecological stability across a wide range of different food-web structures by stabilizing dynamics in some dimensions, while simultaneously destabilizing them in others. Specifically, increasingly positive autocorrelation (reddening) of environmental noise increases resilience by hastening the recovery of food webs following a large perturbation, but reduces their resistance to perturbation and increases their temporal variability (reduces biomass stability). In contrast, all stability dimensions become less predictable, showing increased variability around the mean response, as environmental noise reddens. Moreover, we found environmental reddening to be a considerably more important determinant of stability than intrinsic food-web characteristics. These findings reveal the fundamental and dominant role played by environmental stochasticity in determining the dynamics and stability of ecosystems, and extend our understanding of how the multiple dimensions of stability relate to each other beyond simple white noise environments.