I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.

We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos which admits a cover by the étale spectra of $E_{\infty}$-rings. The definition of a nonconnective spectral *scheme*, on the other hand, is a spectrally-ringed *space* which admits a cover by the Zariski spectra of $E_{\infty}$-rings. Suppose we instead look at spectrally-ringed *topoi* which are locally isomorphic to the Zariski topos of an $E_{\infty}$-ring. Clearly, these generalize nonconnective spectral schemes; and, on the other hand, these objects admit a natural functor to nonconnective spectral DM-stacks, which should be a fully faithful embedding extending the inclusion of schemes as schematic stacks.

My question is this: are these objects actually more general than nonconnective spectral schemes, or is the underlying topos always generated by a topological space? If the former, how much more general are they? (For instance, a nonconnective spectral DM-stack is schematic iff it admits a cover by $(-1)$-truncated affine objects. Is there a similar characterization of these "generalized schematic stacks"?)