For some of us something is not real if it can not be measured (sensed/observed). For others something is real if our best models of the world require its existence. The quantum wave function may then be real. (Such is certainly the case in both the Bohm interpretation of quantum mechanics and in the Everett interpretation and perhaps more generally.) "Real" might best be considered to be a vector (multi dimensional) quantity.
There are even some things like the magnetic vector potential which are perhaps more concrete than the quantum wave function but which (being gauge-dependent) can not be easily measured like the magnetic field itself.
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