  Physicists describe nature using mathematics as the natural language, and for
quantum mechanics, it prefers to use complex numbers. However, whether complex
numbers are really necessary for the theory has been debated ever since its
birth. Recently, it has been shown that a three-party correlation created in
entanglement swapping scenarios comprising independent states and measurements
cannot be reproduced using only real numbers. Previous experiments have
conceptually supported the predication, yet not satisfying the independent
state preparations and measurements simultaneously. Here, we implement such a
test with two truly independent sources delivering entangled photons to three
parties under strict locality conditions. By employing fast quantum random
number generators and high-speed polarization measurements, we space-like
separate all relevant events to ensure independent state preparations and
measurements, and close locality loopholes simultaneously. Our results violate
the real number bound of 7.66 by 5.30 standard deviations, hence rejecting the
universal validity of the real-valued quantum mechanics to describe nature.
