 Net charge from the asymptotic radial electric field

For a spherically symmetric electric field in vacuum, integrate Gauss's law over a sphere. Its electric flux is $4\pi r^2{E}_r(r)$, so taking the enclosing radius to infinity gives the displayed total charge, whenever that limit and total charge exist. In particular, a radial field decaying faster than $r^{-2}$ can be nonzero locally while its total source charge is zero. The origin must also be checked: a point charge contributes a nonzero limiting flux as $r\to 0$.