Edit: I have now found the answer myself, and will write it up nicely on this coming weekend.
A complex Hermitian matrix is unitarily similar to a real symmetric tridiagonal matrix, which one computes as a first step towards full diagonalization, for example using a variation of QR-decomposition using Householder reflections.
But in every source I've found describing reduction of Hermitian matrices to tridiagonal form, it's missing the final step to get a real symmetric tridiagonal matrix from the complex Hermitian tridiagonal matrix that results from the complex Householder reflections. Either the issue is entirely ignored, or it's simply stated that it's possible to make this final transformation. One source even gave it as an exercise for the reader, which is extremely frustrating when that was the single piece of information I was looking for. So now I have to ask: how, concretely, can one compute a unitary transform from Hermitian tridiagonal form to real symmetric tridiagonal form?
I am guessing that something similar to Givens rotations should be used, but I had trouble deriving the equations myself. Any help will be appreciated.
Many thanks in advance.