Spooky action-at-a-distance in general probabilistic theories

We call a probabilistic theory "complete" if it cannot be further refined by no-signaling hidden-variable models, and name a theory "spooky" if every equivalent hidden-variable model violates Shimony's Outcome Independence. We prove that a complete theory is spooky if and only if it admits a pure steering state in the sense of Schrödinger. Finally we show that steering of complementary states leads to a Schrödinger's-cat-like paradox. -- Highlights: ► First analysis of hidden variable models for general probabilistic theories. ► First theory-independent mathematical definition of complementarity. ► Einstein spooky-action-at-distance connected to Schrödinger's pure-state steering. ► Schrödinger's-cat paradox explored outside quantum theory.