  Real eigenvalues of pseudo-Hermitian matrices, such as real matrices and
$\mathcal{PT-}$symmetric matrices, frequently split into complex conjugate
pairs. This is accompanied by the breaking of certain symmetries of the
eigenvectors and, typically, also a drastic change in the behavior of the
system. In this paper, we classify the eigenspace of pseudo-Hermitian matrices
and show that such symmetry breaking occurs if and only if eigenvalues of
opposite kinds collide on the real axis of the complex eigenvalue plane. This
enables a classification of the disconnected regions in parameter space where
all eigenvalues are real -- which correspond, physically, to the stable phases
of the system. These disconnected regions are surrounded by exceptional
surfaces which comprise all the real-valued exceptional points of
pseudo-Hermitian matrices. The exceptional surfaces, together with the diabolic
points created by their intersections, comprise all points of
pseudo-Hermiticity breaking. In particular, this clarifies that the degeneracy
involved in symmetry breaking is not necessarily an exceptional point. We also
discuss how our study relates to conserved quantities and derive the conditions
for when degeneracies caused by external symmetries are susceptible to
thresholdless pseudo-Hermiticity breaking. We illustrate our results with
examples from photonics, condensed matter physics, and mechanics.
