  In quantum mechanics, nonlocality (a violation of a Bell inequality) is
intimately linked to complementarity, by which we mean that consistently
assigning values to different observables at the same time is not possible.
Nonlocality can only occur when some of the relevant observables do not
commute, and this noncommutativity makes the observables complementary. Beyond
quantum mechanics, the concept of complementarity can be formalized in several
distinct ways. Here we describe some of these possible formalizations and ask
how they relate to nonlocality. We partially answer this question by describing
two toy theories which display nonlocality and obey the no-signaling principle,
although each of them does not display a certain kind of complementarity. The
first toy theory has the property that it maximally violates the CHSH
inequality, although the corresponding local observables are pairwise jointly
measurable. The second toy theory also maximally violates the CHSH inequality,
although its state space is classical and all measurements are mutually
nondisturbing: if a measurement sequence contains some measurement twice with
any number of other measurements in between, then these two measurements give
the same outcome with certainty.
