  An operational probabilistic theory where all systems are classical, and all
pure states of composite systems are entangled, is constructed. The theory is
endowed with a rule for composing an arbitrary number of systems, and with a
nontrivial set of transformations. Hence, we demonstrate that the presence of
entanglement is independent of the existence of incompatible measurements. We
then study a variety of phenomena occurring in the theory -- some of them
contradicting both Classical and Quantum Theories -- including: cloning,
entanglement swapping, dense coding, additivity of classical capacities,
non-monogamous entanglement, hypersignaling. We also prove the existence, in
the theory, of a universal processor. The theory is causal and satisfies the
no-restriction hypothesis. At the same time, it violates a number of
information-theoretic principles enjoyed by Quantum Theory, most notably: local
discriminability, purity of parallel composition of states, and purification.
Moreover, we introduce an exhaustive procedure to construct generic operational
probabilistic theories, and a sufficient set of conditions to verify their
consistency. In addition, we prove a characterisation theorem for the parallel
composition rules of arbitrary theories, and specialise it to the case of
bilocal-tomographic theories. We conclude pointing out some open problems. In
particular, on the basis of the fact that every separable state of the theory
is a statistical mixture of entangled states, we formulate a no-go conjecture
for the existence of a local-realistic ontological model.
