  Strong high-ionization iron coronal lines (CLs) are a rare phenomenon
observed in galaxy and quasi-stellar object spectra that are thought to be
created as a result of tidal disruption event (TDE) flares. To test whether
these CLs are the result of TDE activity, we search for extreme coronal line
emitting galaxies (ECLEs) in the Sloan Digital Sky Survey (SDSS), measure their
rate, and compare it to TDE rates from the literature. We detect sufficiently
strong CLs in 14 objects, doubling the number previously found in SDSS. Using
follow-up spectra from the Dark Energy Spectroscopic Instrument and Gemini
Multi-Object Spectrograph, Wide-field Infrared Survey Explorer mid-infrared
observations, and Liverpool Telescope optical photometry, we find that of the
seven new objects, only one evolves in a manner consistent with that of the
five previously discovered variable ECLEs. Using this new sample of six
variable ECLEs, we calculate the galaxy-normalised rate of ECLEs in SDSS to be
$R_\mathrm{G}=2.2~^{+1.3}_{-0.8}~(\mathrm{statistical})~^{+0.0}_{-1.3}~(\mathrm{systematic})\times10^{-5}~\mathrm{galaxy}^{-1}~\mathrm{year}^{-1}$.
The mass-normalised rate is
$R_\mathrm{M}=1.9~^{+1.1}_{-0.7}~(\mathrm{statistical})~^{+0.0}_{-1.1}~(\mathrm{systematic})\times10^{-16}~\mathrm{M_\odot^{-1}}~\mathrm{year}^{-1}$
and the volumetric rate is
$R_\mathrm{V}=6.9~^{+5.6}_{-2.1}~(\mathrm{statistical})~^{+0.0}_{-3.9}~(\mathrm{systematic})\times10^{-8}~\mathrm{Mpc}^{-3}~\mathrm{year}^{-1}$.
Our rates are comparable to TDE rates from the literature, supporting the
suggestion that the CLs in variable ECLEs are the product of TDEs.
