To proposition 5.17 in Weidmann's 'Lineare Operatoren in Hilberträumen' (german version) it is noted that the expansion of compact operators that are normal rather than self adjoint doesn't apply in general for real Hilbert spaces.
Can you give an example where the expansion fails?
One example is given by: $$\pmatrix{0&-1\\1&0}$$
What fails there is that despite being normal (and compact of course) it has no(!) eigenvectors at all. Therefore it cannot be diagonalizable.