We combine for the first time all available information about the spectral shape and morphology of the radio halo of the Coma cluster with the recent γ-ray upper limits obtained by the Fermi-Large Area Telescope (LAT) and with the magnetic field strength derived from Faraday rotation measures. We explore the possibility that the radio emission is due to synchrotron emission of secondary electrons. First, we investigate the case of pure secondary models that are merely based on the mechanism of continuous injection of secondary electrons via proton–proton collisions in the intracluster medium. We use the observed spatial distribution of the halo's radio brightness to constrain the amount of cosmic ray protons and their spatial distribution in the cluster that are required by the model. Under the canonical assumption that the spectrum of cosmic rays is a power law in momentum and that the spectrum of secondaries is stationary, we find that the combination of the steep spectrum of cosmic ray protons necessary to explain the spectrum of the halo and the very broad spatial distribution (and large energy density) of cosmic rays result in a γ-ray emission in excess of present limits, unless the cluster magnetic field is relatively large. However, this large magnetic field required to not violate present γ-ray limits appears inconsistent with that derived from recent Faraday rotation measures. Secondly, we investigate more complex models in which the cosmic rays confined diffusively in the Coma cluster and their secondary electrons are all reaccelerated by magnetohydrodynamics turbulence. We show that under these conditions it is possible to explain the radio spectrum and morphology of the radio halo and to predict γ-ray fluxes in agreement with the Fermi-LAT upper limits without tension with present constraints on the cluster magnetic field. Reacceleration of secondary particles also requires a very broad cosmic ray spatial profile, much flatter than that of the intracluster medium, at least provided that both the turbulent and magnetic field energy densities scale with that of the intracluster medium. However, this requirement can be easily alleviated if we assume that a small amount of (additional) seed primary electrons is reaccelerated in the cluster's external regions, or if we adopt flatter scalings of the turbulent and magnetic field energy densities with distance from the cluster centre.