We characterize entanglement subject to its definition over real and complex, composite quantum systems. In particular, a method is established to assess quantum correlations with respect to a selected number system, illuminating the deeply rooted, yet rarely discussed question of why quantum states are described via complex numbers. With our experiment, we then realize two-photon polarization states that are entangled with respect to the notion of two rebits, comprising two two-level systems over real numbers. At the same time, the generated states are separable with respect to two complex qubits. Among other results, we reconstruct the best approximation of the generated states in terms of a real-valued, local expansion and show that this yields an incomplete description of our data. Conversely, the generated states are shown to be fully decomposable in terms of tensor-product states with complex wave functions. Thereby, we probe paradigms of quantum physics with modern theoretical tools and experimental platforms that are relevant for applications in quantum information science and technology and connected to the fundamentals of the quantum description of nature.
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