  Generically, spectral statistics of spinless systems with time reversal
invariance (TRI) and chaotic dynamics are well described by the Gaussian
Orthogonal ensemble (GOE). However, if an additional symmetry is present, the
spectrum can be split into independent sectors which statistics depend on the
type of the group's irreducible representation. In particular, this allows the
construction of TRI quantum graphs with spectral statistics characteristic of
the Gaussian Symplectic ensembles (GSE). To this end one usually has to use
groups admitting pseudo-real irreducible representations. In this paper we show
how GSE spectral statistics can be realized in TRI systems with simpler
symmetry groups lacking pseudo-real representations. As an application, we
provide a class of quantum graphs with only $C_4$ rotational symmetry
possessing GSE spectral statistics.
