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 Hα 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]/Hα, [N II]/Hα, and [O I]/Hα) with increasing Hα surface brightness (Σ_Hα). 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]/Hβ with Σ_Hα; 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 (LI[N]ERs). Hot low-mass evolved stars make a small contribution to the energy budget of the DIG (2% of the galaxy-integrated Hα 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.