debian+phosh running on PinePhone

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Are you using auto-log-in ?
Seems to me when using GUI, you can make the choice at the log-in screen ...  ?

In Settings, "Automatic Login" is greyed out and cannot be enabled (unless there is an unlock button somewheree off the screen).

Well first off a Wayland implementation of MATE for all distros is probably around the corner (https://www.phoronix.com/scan.php?page=n...-Mir-Video) but probably doesn't work yet

So you'll probably be using Xorg and I'm not sure if there will be missing configurations etc because this version of Debian is purely using Wayland as a display manager (and I do see this as an advantage, Xorg is a nightmare). Other desktop environments are not guaranteed to work, especially ones requiring Xorg, and you'll probably have to do it all yourself. If I was you I would wait for a fully working implementation of MATE for Wayland.

You can at least test out other desktop environments by installing them and also gdm, then you would just log out and be taken to gdm's login manager. At the login manager is where you would change the selected desktop.