simulated stellar populations share the same velocity profiles and exhibit the same errors as the observed stars. For each bin we calculated the observed difference and counted the number of simulations that resulted in a difference greater than the observed one.

The fraction of simulations, $N^{*}/N_{\rm sim}$, where $\Delta_{\rm chi}\geq \Delta_{\rm obs}$ is indicative of the significance of the difference of the observed profiles. To quantify the global significance we computed the median of the significance of each bin, as well as the maximum and minimum values along the observed profiles. The results are listed in Table \[tab:Significance\] for each pair of velocity profiles.

The results from the Anderson-Darling test and the values of $N_{\rm sim}$ are listed in Table \[tab:Significance\] and reveal that we find no differences between observed velocity profiles of Fe-poor and Fe-rich stars, neither in $\omega\,$Centauri nor in M22.

### Global rotation

To further quantify the rotation in the plane of the sky of both clusters we adopted the procedure described in @cordoni2019 [see their Section 3].

Briefly, we divided the field of view of each cluster into 30 circular sectors with an arc length of 45$^{\circ}$. We computed the median $\mu_{\rm \alpha}\cos\delta$ and $\mu_{\rm \delta}$ 