There are many cases, for simplicity, we choose the wavefunctions to be real. For example, in http://en.wikipedia.org/wiki/Born–Oppenheimer_approximation, there is "The electronic wave functions \chi_k\, will be taken to be real, which is possible when there are no magnetic or spin interactions. " I do not know why this can always be done. In fact, I think even the electron eigenstates of hydrogen cannot taken to be real. I know if the Hamiltonian has time-reversal symmetry, and the energy eigenstate is nondegenerate, then the wavefunction can taken to be real. But in most cases there is degeneracy. Therefore this assumption seems to be starnge and not reasonable?