In Griffiths Introduction to Quantum Mechanics (3ed.) problem 2.1, we are asked to prove that the normalizable solutions to the time-independent Schrödinger equation can always be chosen to be real, in a simple 1-dimensional position-space setup. The question instructs the reader to do this by first showing the complex conjugate of a solution is guaranteed to be a solution, and then using this to obtain purely real solutions by linearity.
What I'm wondering is whether there is a way to generalise this to any Hamiltonian and any Hilbert space, or a certain subset thereof. The statement would be something like “in any Hilbert space with property X and any Hermitian operator $\hat H$ with property Y it is possible to give a complete set of eigenvectors of $\hat H$ with only real coëfficients expressed in a basis with property Z”.
If this doesn't hold generally, what's special about position-space that allows one to do this?
