The Auger-TA composition working group reports on a comparison of $X_\mathrm{max}$ distributions measured by the Pierre Auger and Telescope Array Observatories. The shapes of the $X_\mathrm{max}$ distributions measured by the Auger and TA Observatories are evaluated and a quantitative compatibility test is presented. A direct comparison of the measured $X_\mathrm{max}$ distributions is not correct due to different detector acceptances and resolutions as well as different analysis techniques. In this contribution, a method developed to allow a correct comparison of the $X_\mathrm{max}$ distributions is explained and used. A set of showers compatible to the composition measured by the Auger detectors was simulated and reconstructed using the official TA software chain. This procedure simulates an energy-dependent composition mixture, which represents a good fit to Auger $X_\mathrm{max}$ distributions, exposed through the detector acceptance, resolution and analysis procedure of the TA experiment. Two compatibility tests are applied to the $X_\mathrm{max}$ distributions: Kolmogorov-Smirnov and Anderson-Darling. Both tests shows that TA data is within the systematic uncertainties compatible to a mixed composition such as the one measured by the Auger detectors