  The synthetic analogue functor $\nu$ from spectra to synthetic spectra does
not preserve all limits. In this paper, we give a necessary and sufficient
criterion for $\nu$ to preserve the global sections of a derived stack. Even
when these conditions are not satisfied, our framework still yields synthetic
spectra that implement the descent spectral sequence for the structure sheaf,
thus placing descent spectral sequences on good footing in the
$\infty$-category of synthetic spectra. As an example, we introduce a new
$\mathrm{MU}$-synthetic spectrum $\mathrm{Smf}$.
