The role of complex quantities in quantum theory has been puzzling physicists since the beginnings. It is, thus, natural to ask whether, in order to describe our experiments, the mathematical structure of the complex Hilbert spaces it is built on is really necessary. Recently, it was shown that this structure is inevitable in network scenarios with independent sources. More precisely, Real Quantum Theory cannot explain the predictions of (Complex) Quantum Theory [Renou et al., Nature (London) 600, 625 (2021).]. Here, we revisit the independence assumption underlying this work. We show that assuming partial independence is sufficient for showing the inadequacy of Real Quantum Theory. We derive a tradeoff between source independence and the Bell value achievable in Real Quantum Theory, which also lower bounds the source correlations required to explain previous experiments by means of real quantum systems. We further show that 1 bit of entanglement is necessary and sufficient for recovering the complex quantum correlations by means of Real Quantum Theory in the scenario from [Renou et al., Nature (London) 600, 625 (2021).]. Finally, building on [McKague et al., Phys. Rev. Lett. 102, 2009], we provide a construction to simulate any complex quantum setup with $m$ independent sources by means of Real Quantum Theory by allowing the sources to share an $m$ real-qubit entangled state in the first round of the experiment.

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• Nonlocality
• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism
• Quantum foundations