Having thought about it a little more, this is what I come up with:
Newton's third law says that oscillators i and j exert equal
magnitude forces on each other. This in turn will lead to elements
K_ij and K_ji in the force-constant matrix K to be equal. In turn,
this means K is symmetric (and of course it's real); and so is the
mass matrix M. But this means the eigenvectors must be real, even if
I write the oscillator displacement vector x in complex form where
the i-th component is A_i exp (i d_i). But this means d_i can only be
0 or pi.