@vv For a system where all the parameters are within a few orders of magnitude of unity, it makes sense to interpret something like 10^(-18) as just a numerical zero and not significant. However, here the equilibrium constants are very far from 1, and so we expect some concentrations to be very small. So for a well-posed physical problem, there will be a unique physical solution with positive concentrations among other nonphysical solutions with some negative concentrations.
For the three independent reactions here if we neglect interactions betweem molecules (set all the u_X variables to 1), then the first 6 equations will have a unique physical solution in which we expect the concentrations of H+ and OH- and the ion-pair HCl to be very small.
The other equations are about the interactions between molecules and mean that the u_X quantities depart from 1, but not by that much - see the plot. These are empirical equations and need not be physically self-consistent, and can lead to there being no solution at all in the physical regime. On the other hand if the solution is comparable to the one for eqns 1-6 with the u_X =1, which seems to be the case here though I didn't check it quantitatively, then there is good reason to think that the solution is physical and that the small positive concentrations are not just numerical zeroes.