We present a comparison of existing experimental data for the radiative leptonic decays $Ptoell u_ellgamma$, where $P=K$ or $pi$ and $ell=e$ or $mu$, from the KLOE, PIBETA, E787, ISTRA+ and OKA collaborations with theoretical predictions based on the recent non-perturbative determinations of the structure-dependent vector and axial-vector form factors, $F_V$ and $F_A$ respectively. These were obtained using lattice QCD+QED simulations at order $O(alpha_{mathrm{em}})$ in the electromagnetic coupling. We find good agreement with the KLOE data on $Kto e u_egamma$ decays from which the form factor $F^+=F_V+F_A$ can be determined. For $Ktomu u_mugamma$ decays we observe differences of up to 3,-,4 standard deviations at large photon energies between the theoretical predictions and the data from the E787, ISTRA+ and OKA experiments and similar discrepancies in some kinematical regions with the PIBETA experiment on radiative pion decays. A global study of all the kaon-decay data within the Standard Model results in a poor fit, largely because at large photon energies the KLOE and E787 data cannot be reproduced simultaneously in terms of the same form factor $F^+$. The discrepancy between the theoretical and experimental values of the form factor $F^-=F_V-F_A$ is even more pronounced. These observations motivate future improvements of both the theoretical and experimental determinations of the structure-dependent form factors $F^+$ and $F^-$, as well as further theoretical investigations of models of new physics which might for example, include possible flavor changing interactions beyond $V - A$ and/or non-universal corrections to the lepton couplings.
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