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Abstract Earth's tropics are characterized by quasi‐steady precipitation with small oscillations about a mean value, which has led to the hypothesis that moist convection is in a state of quasi‐equilibrium (QE). In contrast, very warm simulations of Earth's tropical convection are characterized by relaxation‐oscillator‐like (RO) precipitation, with short‐lived convective storms and torrential rainfall forming and dissipating at regular intervals with little to no precipitation in between. We develop a model of moist convection by combining a zero‐buoyancy model of bulk‐plume convection with a QE heat engine model, and we use it to show that QE is violated at high surface temperatures. We hypothesize that the RO state emerges when the equilibrium condition of the convective heat engine is violated, that is, when the heating rate times a thermodynamic efficiency exceeds the rate at which work can be performed. We test our hypothesis against one‐ and three‐dimensional numerical simulations and find that it accurately predicts the onset of RO convection. The proposed mechanism for RO emergence from QE breakdown is agnostic of the condensable, and can be applied to any planetary atmosphere undergoing moist convection. To date, RO states have only been demonstrated in three‐dimensional convection‐resolving simulations, which has made it seem that the physics of the RO state requires simulations that can explicitly resolve the three‐dimensional interaction of cloudy plumes and their environment. We demonstrate that RO states also exist in single‐column simulations of radiative‐convective equilibrium with parameterized convection, albeit in a different surface temperature range and with much longer storm‐free intervals.