We have explored the capabilities of dust extinction and $gamma$ rays to probe the properties of the interstellar medium in the nearby anti-centre region. We have jointly modelled the $gamma$-ray intensity and the stellar reddening, E(B-V) as a combination of H$_{rm I}$-bright, CO-bright, and ionised gas components. The complementary information from dust reddening and $gamma$ rays is used to reveal the dark gas not seen, or poorly traced, by H$_{rm I}$, free-free, and $^{12}$CO emissions. We compare the total gas column densities, $N_{rm{H}}$, derived from the $gamma$ rays and stellar reddening with those inferred from a similar analysis (Remy et al. 2017) of $gamma$ rays and of the optical depth of the thermal dust emission, $tau_{353}$, at 353 GHz. We can therefore compare environmental variations in specific dust reddening, E(B-V)/$N_{rm H}$, and in dust emission opacity (dust optical depth per gas nucleon), $tau_{353}/N_{rm{H}}$. Over the whole anti-centre region, we find an average E(B-V)/$N_{rm H}$ ratio of $(2.02pm0.48)times$ $10^{-22}$~mag~cm$^2$, with maximum local variations of about $pm30%$ at variance with the two to six fold coincident increase seen in emission opacity as the gas column density increases. In the diffuse medium, the small variations in specific reddening, E(B-V)/$N_{rm H}$ implies a rather uniform dust-to-gas mass ratio in the diffuse parts of the anti-centre clouds. The small amplitude of the E(B-V)/$N_{rm H}$ variations with increasing $N_{rm{H}}$ column density confirms that the large opacity $tau_{353}/N_{rm{H}}$ rise seen toward dense CO clouds is primarily due to changes in dust emissivity. The environmental changes are qualitatively compatible with model predictions based on mantle accretion on the grains and the formation of grain aggregates.
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