  Quantum theory is commonly formulated in complex Hilbert spaces. However, the
question of whether complex numbers need to be given a fundamental role in the
theory has been debated since its pioneering days. Recently it has been shown
that tests in the spirit of a Bell inequality can reveal quantum predictions in
entanglement swapping scenarios that cannot be modelled by the natural
real-number analog of standard quantum theory. Here, we tailor such tests for
implementation in state-of-the-art photonic systems. We experimentally
demonstrate quantum correlations in a network of three parties and two
independent EPR sources that violate the constraints of real quantum theory by
over $4.5$ standard deviations, hence disproving real quantum theory as a
universal physical theory.
