  The question of whether complex numbers play a fundamental role in quantum
theory has been debated since the inception of quantum mechanics. Recently, a
feasible proposal to differentiate between real and complex quantum theories
based on the technique of testing Bell nonlocalities has emerged [Nature 600,
625-629 (2021)]. Based on this method, the real quantum theory has been
falsified experimentally in both photonic and superconducting quantum systems
[Phys. Rev. Lett. 128, 040402 (2022), Phys. Rev. Lett. 128, 040403 (2022)]. The
quantum networks with multiple independent sources which are not causally
connected have gained significant interest as they offer new perspective on
studying the nonlocalities. The independence of these sources imposes
additional constraints on observable covariances and leads to new bounds for
classical and quantum correlations. In this study, we examine the
discrimination between the real and complex quantum theories with an
entanglement swapping scenario under a stronger assumption that the two sources
are causally independent, which wasn't made in previous works. Using a revised
Navascu\'es-Pironio-Ac\'in method and Bayesian optimization, we find a proposal
with optimal coefficients of the correlation function which could give a larger
discrimination between the real and quantum theories comparing with the
existing proposals. This work opens up avenues for further exploration of the
discrimination between real and complex quantum theories within intricate
quantum networks featuring causally independent parties.
