Model predictions for the [O III]/Hβ line ratio considering both young stars and HOLMES. a) Model predictions for photoionised clouds, computed with CLOUDY and a 2 Myr (solid lines) and a 10 Gyr (dashed lines) SSP spectrum. The line ratio log([O III]/Hβ) is plotted as a function of the ionisation parameter at the incident face of the cloud (log(U₀)). The vertical extent of the grey area highlights the range of line ratios observed in the DIG in the PHANGS–MUSE data. Different colours correspond to different gas-phase metallicities, with the same colour-coding in both panels, as indicated in the legend. All models predict log([O III]/Hβ) to increase with Σ_Hα, in contrast to what is observed. b) Model predictions for the log([O III]/Hβ) line ratio as a function of the fraction of the ionising radiation originating from old stars, f_ion = Φ_OB/Φ_HOLMES. We show models for a fixed value of log(U₀) = − 3.5 and different metallicities. The horizontal light grey band corresponds to the range of values of log([O III]/Hβ) observed in the DIG of our sample, while the dark grey box corresponds to the range of f_ion observed in the PHANGS data. f_ion is estimated by computing the ionising photons expected to be produced by HOLMES given the local stellar mass surface density.