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We combine results from deep ALMA observations of massive($M_*>10^{10}\;M_{\odot}$) galaxies at different redshifts to show that thecolumn density of their inter stellar medium (ISM) rapidly increases towardsearly cosmic epochs. Our analysis includes objects from the ASPECS and ALPINElarge programs, as well as individual observations of $z\sim 6$ QSO hosts. Whenaccounting for non-detections and correcting for selection effects, we findthat the median surface density of the ISM of the massive galaxy populationevolves as $\sim(1+z)^{3.3}$. This means that the ISM column density towardsthe 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 fromX-ray observations of large AGN samples already at $z\gtrsim2$.  We develop a simple analytic model for the spatial distribution of ISM cloudswithin galaxies, and estimate the total covering factor towards active nucleiwhen obscuration by ISM clouds on the host scale is added to that of pc-scalecircumnuclear material (the so-called 'torus'). The model includes clouds witha distribution of sizes, masses, and surface densities, and also allows for anevolution of the characteristic cloud surface density with redshift,$\Sigma_{c,*}\propto(1+z)^\gamma$. We show that, for $\gamma=2$, such a modelsuccessfully reproduces the increase of the obscured AGN fraction with redshiftthat is commonly observed in deep X-ray surveys, both when different absorptionthresholds and AGN luminosities are considered.  Our results suggest that 80-90\% of supermassive black holes in the earlyUniverse ($z>6-8$) are hidden to our view, primarily by the ISM in their hosts.[abridged]

Supermassive Black Holes at High Redshift are Expected to be Obscured by their Massive Host Galaxies' Inter Stellar Medium