  We use integral field spectroscopy from the PHANGS-MUSE survey, which
resolves the ionised interstellar medium at ${\sim}50$ 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 HII regions and then binning the low-surface-brightness
areas to achieve significant detections of the key nebular lines. A simple
model for the leakage and propagation of ionising radiation from HII regions is
able to reproduce the observed distribution of H$\alpha$ in the DIG. Leaking
radiation from HII regions also explains the observed decrease in line ratios
of low-ionisation species ([SII]/H$\alpha$, [NII]/H$\alpha$ and [OI]/H$\alpha$)
with increasing H$\alpha$ surface brightness ($\Sigma_{H\alpha}$). 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
galaxies; (2) the observed trends of a flat or decreasing [OIII]/H$\beta$ with
$\Sigma_{H\alpha}$; and (3) the offset of some DIG regions from the locus of
HII 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 H$\alpha$ emission), but their harder spectra make them
fundamental contributors to [OIII] 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.
