Using model simulations, we demonstrate that the response of top-of-atmosphere radiative fluxes to localized tropical sea surface temperature (SST) perturbations exhibits numerous non-linearities. Most pronounced is an ‘asymmetry’ in the response to positive and negative SST perturbations. Additionally, we identify a ‘magnitude-dependence’ of response on the size of the SST perturbation. We then explain how these non-linearities arise as a robust consequence of convective quasi-equilibrium and weak (but non-zero) temperature gradients in the tropical free-troposphere, which we encapsulate in a ‘circus tent’ model of the tropical atmosphere. These results demonstrate that the climate response to SST perturbations is fundamentally non-linear, and highlight potential deficiencies in work which has assumed linearity in the response.