  Complex numbers, i.e., numbers with a real and an imaginary part, are
essential for mathematical analysis, while their role in other subjects, such
as electromagnetism or special relativity, is far less fundamental. Quantum
physics is the only physical theory where these numbers seem to play an
indispensible role, as the theory is explicitly formulated in terms of
operators acting on complex Hilbert spaces. The occurrence of complex numbers
within the quantum formalism has nonetheless puzzled countless physicists,
including the fathers of the theory, for whom a real version of quantum
physics, where states and observables are represented by real operators, seemed
much more natural. In fact, previous works showed that such "real quantum
physics" can reproduce the outcomes of any multipartite experiment, as long as
the parts share arbitrary real quantum states. Thus, are complex numbers really
needed for a quantum description of nature? Here, we show this to be case by
proving that real and complex quantum physics make different predictions in
network scenarios comprising independent quantum state sources. This allows us
to devise a Bell-type quantum experiment whose input-output correlations cannot
be approximated by any real quantum model. The successful realization of such
an experiment would disprove real quantum physics, in the same way as standard
Bell experiments disproved local physics.
