  Recent investigations of the pseudospectrum in black hole spacetimes have
shown that quasinormal mode frequencies suffer from spectral instabilities.
This phenomenon may severely affect gravitational-wave spectroscopy and limit
precision tests of general relativity. We extend the pseudospectrum analysis to
horizonless exotic compact objects which possess a reflective surface
arbitrarily close to the Schwarzschild radius, and find that their quasinormal
modes also suffer from an overall spectral instability. Even though all the
modes themselves decay monotonically, the pseudospectrum contours of equal
resonance magnitude around the fundamental mode and the lowest overtones can
cross the real axis into the unstable regime of the complex plane, unveiling
the existence of nonmodal pseudo-resonances. A pseudospectrum analysis further
predicts that fluctuations to the system may destabilize the object when next
to leading-order effects are considered, as the triggering of pseudo-resonant
growth can break the order-expansion of black-hole perturbation theory.
