  We introduce a new class of non-compact backgrounds of Type IIB string theory
preserving eight supercharges by combining S-folds and non-perturbative
7-branes wrapping orbifolds, and study the four-dimensional superconformal
field theories arising at low energy on $D3$-branes probing them. We draw a
precise correspondence between this setup and the torus compactification of
six-dimensional orbi-instanton theories with a Stiefel-Whitney twist, and use
it to determine the main features of such strongly-coupled systems, like
central charges, spectra of Coulomb-branch operators, networks of Higgs-branch
flows. Finally, with the aim to improve our understanding of the landscape of
$\mathcal{N}=2$ superconformal field theories, and possibly to extend their
classification beyond rank two, we provide a detailed catalogue of all the
rank-three theories that our framework gives access to.
