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H$_{\rm I}$ 21-cm and $^{12}$CO 2.6-mm line emissions respectively trace theatomic and molecular gas phases, but they miss most of the opaque H$_{\rm I}$and diffuse H$_2$ present in the Dark Neutral Medium (DNM) at the transitionbetween the H$_{\rm I}$-bright and CO-bright regions. Jointly probing H$_{\rmI}$, CO, and DNM gas, we aim to constrain the threshold of the H$_{\rmI}$-H$_2$ transition. We also aim to measure gas mass fractions in thedifferent phases and to test their relation to cloud properties. We have useddust optical depth measurements at 353 GHz, $\gamma$-ray maps at GeV energies,and H$_{\rm I}$ and CO line data to trace the gas column densities and map theDNM in nearby clouds toward the Galactic anticentre and Chamaeleon regions. Theonset of detectable CO intensities varies by only a factor of 4 from cloud tocloud, between 0.6$ \times 10^{21}$ cm$^{-2}$ and 2.5$ \times 10^{21}$cm$^{-2}$ in total gas column density. We observe larger H$_2$ column densitiesthan linearly inferred from the CO intensities at $A_{\rm{V}}> 3$ magnitudesbecause of the large CO optical thickness: the additional H$_2$ mass in thisregime represents on average 20$\%$ of the CO-inferred molecular mass. In theDNM envelopes, we find that the fraction of diffuse CO-dark H$_2$ in themolecular column densities decreases with increasing $A_{\rm{V}}$ in a cloud.For a half molecular DNM, the fraction decreases from more than 80$\%$ at 0.4magnitude to less than 20$\%$ beyond 2 magnitudes. In mass, the DNM fractionvaries with the cloud properties. The mass present in the DNM envelopes appearsto scale with the molecular mass seen in CO as $M_{\rm H}^{\rm DNM} = (62 \pm7) {M_{\rm H_2}^{\rm CO}}^{\,0.51 \pm 0.02}$ across two decades in mass. Thephase transitions in these clouds show both common trends and environmentaldifferences.

Cosmic-rays, gas, and dust in nearby anticentre clouds : II -- Interstellar phase transitions and the Dark Neutral Medium