Many theories of nonlinear electrodynamics (NLED) that have been proposed in physical contexts involving strong fields are causal for weak fields but acausal for strong fields. We show that for any such theory there is a unique causal and self-dual (electromagnetic duality invariant) theory with the same Lagrangian at zero magnetic field. This follows from a construction of the general causal self-dual NLED, which shows that strong-field causality is implied by weak-field causality for self-dual theories. We illustrate our results with explicit examples. Published by the American Physical Society 2024