We use integral field spectroscopy from the PHANGS–MUSE survey, which resolves the ionised interstellar medium structure at ∼50 pc resolution in 19 nearby spiral galaxies, to study the origin of the diffuse ionised gas (DIG). We examine the physical conditions of the diffuse gas by first removing morphologically defined H  II regions and then binning the low-surface-brightness areas to achieve significant detections of the key nebular lines in the DIG. A simple model for the leakage and propagation of ionising radiation from H  II regions is able to reproduce the observed distribution of H α in the DIG. This model infers a typical mean free path for the ionising radiation of 1.9 kpc for photons propagating within the disc plane. Leaking radiation from H  II regions also explains the observed decrease in line ratios of low-ionisation species (S  II /H α , N  II /H α , and O  I /H α ) with increasing H α surface brightness (Σ H α ). Emission from hot low-mass evolved stars, however, is required to explain: (1) the enhanced low-ionisation line ratios observed in the central regions of some of the galaxies in our sample; (2) the observed trends of a flat or decreasing O  III /H β with Σ H α ; and (3) the offset of some DIG regions from the typical locus of H  II regions in the Baldwin–Phillips–Terlevich (BPT) diagram, extending into the area of low-ionisation (nuclear) emission-line regions (LINERs). Hot low-mass evolved stars make a small contribution to the energy budget of the DIG (2% of the galaxy-integrated H α emission), but their harder spectra make them fundamental contributors to O  III emission. The DIG might result from a superposition of two components, an energetically dominant contribution from young stars and a more diffuse background of harder ionising photons from old stars. This unified framework bridges observations of the Milky Way DIG with LI(N)ER-like emission observed in nearby galaxy bulges.