  Ernst Specker considered a particular feature of quantum theory to be
especially fundamental, namely that pairwise joint measurability of sharp
measurements implies their global joint measurability
(https://vimeo.com/52923835). To date, Specker's principle seemed incapable of
singling out quantum theory from the space of all general probabilistic
theories. In particular, its well-known consequence for experimental
statistics, the principle of consistent exclusivity, does not rule out the set
of correlations known as almost quantum, which is strictly larger than the set
of quantum correlations. Here we show that, contrary to the popular belief,
Specker's principle cannot be satisfied in any theory that yields almost
quantum correlations.
