This study investigates the mechanisms by which small-scale turbulence and cloud physics determine the organization of large-scale convection in radiative-convective equilibrium (RCE), an idealization of the tropical atmosphere. Under uniform forcings similar to typical tropical conditions, the atmosphere in RCE might spontaneously separate into dry and moist regions on scales of 100-1000 km, with convective clouds aggregating into a cluster in the latter. This phenomenon is known as convective self-aggregation. Herein, we demonstrate that subtle changes in assumptions related to cloud physics and turbulence on scales of ~1 km can dictate the emergence or suppression of convective self-aggregation, resulting from a bifurcation of the dynamical system. The bifurcation occurs when a small dry patch forms in the domain and is sustained because it contributes to negative effective diffusivity of the circulation. Cloud-radiation feedbacks and turbulence circulation interactions govern the formation of such dry patches, thereby modulating the bifurcation. This sensitive dependence on subgrid process models might be a fundamental barrier to climate predictability in light of inherent uncertainties in microscale processes. Because without the capability to include exact representations of those processes in climate models, slight differences in the different approximations used by modelers can lead to qualitative changes in climate predictions, at least for some processes.