  In the past five years, six quasi-periodic X-ray eruption (QPE) sources have
been discovered in the nuclei of nearby galaxies. Their origin remains an open
question. We present MUSE integral field spectroscopy of five QPE host galaxies
to characterize their properties. We find that 3/5 galaxies host extended
emission line regions (EELRs) up to 10 kpc in size. The EELRs are photo-ionized
by a non-stellar continuum, but the current nuclear luminosity is insufficient
to power the observed emission lines. The EELRs are decoupled from the stars
both kinematically and in projected sky position, and the low velocities and
velocity dispersions ($<$ 100 km s$^{-1}$ and $\lesssim 75$ km s$^{-1}$
respectively) are inconsistent with being AGN- or shock-driven. The origin of
the EELRs is likely a previous phase of nuclear activity. The QPE host galaxy
properties are strikingly similar to those of tidal disruption events (Wevers
et al. submitted). The preference for a very short-lived (the typical EELR
lifetime is $\sim$15000 years), gas-rich phase where the nucleus has recently
faded significantly suggests that TDEs and QPEs may share a common formation
channel, disfavoring AGN accretion disk instabilities as the origin of QPEs. In
the assumption that QPEs are related to extreme mass ratio inspiral systems
(EMRIs; stellar-mass objects on bound orbits about massive black holes), the
high incidence of EELRs and recently faded nuclear activity can be used to aid
in the localization of the host galaxies of EMRIs discovered by low frequency
gravitational wave observatories.
