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 [OI]/Hα) with increasing Hα surface brightness (ΣHα). 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 ΣHα; 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.