  We study the full phase diagram of a non-Hermitian PT-symmetric
generalization of the paradigmatic two-dimensional massive Thirring model.
Employing the non-perturbative functional renormalization-group, we find that
the model hosts a regime where PT symmetry is spontaneously broken. This new
phase is characterized by a relevant imaginary mass, corresponding to
monstronic excitations displaying exponentially growing amplitudes for
time-like intervals and tachyonic (Lieb-Robison-bound breaking, oscillatory)
excitations for space-like intervals. Furthermore, since the phase manifests
itself as an unconventional attractive spinodal fixed point, which is typically
unreachable in finite real-life systems, we find that the effective
renormalized mass reached can be tuned through the microscopic parameters of
the model. Our results further predict that the new phase is robust to external
gauge fields, contrary to the celebrated BKT phase in the PT unbroken sector.
The gauge field then provides an effective and easy means to tune the
renormalized imaginary mass through a wide range of values, and therefore the
amplitude growth/oscillation rate of the corresponding excitations.
