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We provide a dispersion-theoretical representation of the reaction amplitudes$\gamma K\to K \pi$ in all charge channels, based on modern pion-kaon $P$-wavephase shift input. Crossed-channel singularities are fixed from phenomenologyas far as possible. We demonstrate how the subtraction constants can be matchedto a low-energy theorem and radiative couplings of the $K^*(892)$ resonances,thereby providing a model-independent framework for future analyses ofhigh-precision kaon Primakoff data.

Dispersive analysis of the Primakoff reaction $\gamma K \to K \pi$