  We study local-global principles for two notions of semi-integral points,
termed Campana points and Darmon points. In particular, we develop a
semi-integral version of the Brauer-Manin obstruction interpolating between
Manin's classical version for rational points and the integral version
developed by Colliot-Th\'el\`ene and Xu. We determine the status of
local-global principles, and obstructions to them, in two families of orbifolds
naturally associated to quadric hypersurfaces. Further, we establish a
quantitative result measuring the failure of the semi-integral Brauer-Manin
obstruction to account for its integral counterpart for affine quadrics.
