You're right. Such a thing is not expressible as a wavefunction, as an unentangled pure state. Entanglement with something else that you are ignorant of (unentangled with) is one way of getting the right statistics. So too is expressing it as an impure state in the density matrix formalism. Some people appeal to the "church of the larger Hilbert space", saying that only pure states exist, and that system is entangled with other, unobservable ones.