  At the Pierre Auger Observatory and the Telescope Array, the measurements of
depths of maximum of air-shower profiles, $X_\mathrm{max}$, are performed using
direct observations of the longitudinal development of showers with the help of
the fluorescence telescopes. Though the same detection technique is used at
both installations, the straightforward comparison of the characteristics of
the measured $X_\mathrm{max}$ distributions is not possible due to the
different approaches to the analysis of the recorded events. In this work, the
Auger-Telescope Array composition working group presents a technique to compare
the $X_\mathrm{max}$ measurements from the Auger Observatory and the Telescope
Array. Applying this technique the compatibility of the first two moments of
the measured $X_\mathrm{max}$ distributions is qualitatively tested for
energies $10^{18.2}~{\rm eV} < E < 10^{19.0}~{\rm eV}$ using the recently
published Telescope Array data from the Black Rock Mesa and Long Ridge
fluorescence detector stations. For a quantitative comparison, simulations of
air showers with EPOS-LHC, folded with effects of the Telescope Array detector,
are required along with the inclusion in the analysis of the systematic
uncertainties in the measurements of $X_\mathrm{max}$ and the energies of the
events.
