  We contrast the latest observations of the cosmic metal density in neutral
gas ($\rho_{\rm met,neu}$) with three cosmological galaxy evolution
simulations: L-GALAXIES 2020, TNG100, and EAGLE. We find that the fraction of
total metals that are in neutral gas is $<40$ per cent at $3\lesssim{} z
\lesssim{} 5$ in these simulations, whereas observations of damped
Lyman-$\alpha$ (DLA) systems suggest $\gtrsim{}85$ per cent. In all three
simulations, hot, low-density gas is also a major contributor to the cosmic
metal budget, even at high redshift. By considering the evolution in cosmic SFR
density ($\rho_{\rm SFR}$), neutral gas density ($\rho_{\rm HI}$), and mean
gas-phase metallicity ($[\langle{}{\rm M/H}\rangle{}]_{\rm neu}$), we determine
two possible ways in which the $\rho_{\rm met,neu}$ observed in DLAs at high
redshift can be matched by simulations: (a) the $\rho_{\rm SFR}$ at
$z\gtrsim{}3$ is greater than inferred from current FUV observations, or (b)
current high-redshift DLA metallicity samples have a higher mean host mass than
the overall galaxy population. If the first is correct, TNG100 would match the
ensemble data best, however there would be an outstanding tension between the
currently observed $\rho_{\rm SFR}$ and $\rho_{\rm met,neu}$. If the second is
correct, L-GALAXIES 2020 would match the ensemble data best, but would require
an increase in neutral gas mass inside subhaloes above $z\sim{}2.5$. If neither
is correct, EAGLE would match the ensemble data best, although at the expense
of over-estimating $[\langle{}{\rm M/H}\rangle{}]_{\rm neu}$. Modulo details
related to numerical resolution and HI mass modelling in simulations, these
incompatibilities highlight current tensions between key observed cosmic
properties at high redshift.
