Complex numbers are essential in mathematics, but they are not needed to describe physical experiments, expressed in terms of probabilities, hence real numbers. Physics however aims to explain, rather than describe, experiments through theories. While most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces. However, could a real version of quantum theory, in terms of real operators, lead to the same predictions as  standard quantum theory? In fact, previous works showed that such “real quantum theory” can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states. Are therefore complex numbers really needed in the standard Hilbert space formulation of quantum theory? In the talk, we provide an answer to this question.