  Studies of many-body non-Hermitian parity-time (PT)-symmetric quantum systems
are attracting a lot of interest due to their relevance in research areas
ranging from quantum optics and continuously monitored dynamics to Euclidean
wormholes in quantum gravity and dissipative quantum chaos. While a symmetry
classification of non-Hermitian systems leads to 38 universality classes, we
show that, under certain conditions, PT-symmetric systems are grouped into 24
universality classes. We identify 14 of them in a coupled two-site
Sachdev-Ye-Kitaev (SYK) model and confirm the classification by spectral
analysis using exact diagonalization techniques. Intriguingly, in 4 of these 14
universality classes, AIII$_\nu$, BDI$^\dagger_\nu$, BDI$_{++\nu}$, and
CI$_{--\nu}$, we identify a basis in which the SYK Hamiltonian has a block
structure in which some blocks are rectangular, with $\nu \in \mathbb{N}$ the
difference between the number of rows and columns. We show analytically that
this feature leads to the existence of $\nu$ robust purely \emph{real}
eigenvalues, whose level statistics follow the predictions of Hermitian random
matrix theory for classes A, AI, BDI, and CI, respectively. We have recently
found that this $\nu$ is a topological invariant, so these classes are
topological. By contrast, nontopological real eigenvalues display a crossover
between Hermitian and non-Hermitian level statistics. Similarly to the case of
Lindbladian dynamics, the reduction of universality classes leads to unexpected
results, such as the absence of Kramers degeneracy in a given sector of the
theory. Another novel feature of the classification scheme is that different
sectors of the PT-symmetric Hamiltonian may have different symmetries.
