Supermassive black holes at high redshift are expected to be obscured by their massive host galaxies' interstellar medium

We combine results from deep ALMA observations of massive ($M_*>10^{10}\;M_{\odot}$) galaxies at different redshifts to show that the column density of their inter stellar medium (ISM) rapidly increases towards early cosmic epochs. Our analysis includes objects from the ASPECS and ALPINE large programs, as well as individual observations of $z\sim 6$ QSO hosts. When accounting for non-detections and correcting for selection effects, we find that the median surface density of the ISM of the massive galaxy population evolves as $\sim(1+z)^{3.3}$. This means that the ISM column density towards the nucleus of a $z>3$ galaxy is typically $>100$ times larger than locally, and it may reach values as high as Compton-thick at $z\gtrsim6$. Remarkably, the median ISM column density is of the same order of what is measured from X-ray observations of large AGN samples already at $z\gtrsim2$. We develop a simple analytic model for the spatial distribution of ISM clouds within galaxies, and estimate the total covering factor towards active nuclei when obscuration by ISM clouds on the host scale is added to that of pc-scale circumnuclear material (the so-called 'torus'). The model includes clouds with a distribution of sizes, masses, and surface densities, and also allows for an evolution of the characteristic cloud surface density with redshift, $\Sigma_{c,*}\propto(1+z)^\gamma$. We show that, for $\gamma=2$, such a model successfully reproduces the increase of the obscured AGN fraction with redshift that is commonly observed in deep X-ray surveys, both when different absorption thresholds and AGN luminosities are considered. Our results suggest that 80-90\% of supermassive black holes in the early Universe ($z>6-8$) are hidden to our view, primarily by the ISM in their hosts. [abridged]