Initially, full screen doesn't pop up - after alt-tabbing, the problem seems to be fixed.

After following Leon's advice (in the Tribes of the East protondb section), everything appears to be working fine. As per his report:

"Here are my launch parameters: "PROTON_NO_D3D11=1 PROTON_NO_ESYNC=1 PROTON_USE_WINED3D=1 %command%".

Game launched successfully, but the cursor stuck within an invisible borders and I could not select any menu item. The solution was to change cursor type to hw. Config file located here: /SteamLibrary/steamapps/compatdata/15370/pfx/drive_c/users/steamuser/My Documents/My Games/Heroes of Might and Magic V - Tribes of the East/Profiles//user_a2.cfg Need to change setvar ui_use_hwcursor from 0 to 1."

I also had some slowdown issues when I started the game with 'Very high' graphics. By setting it to 'Normal', and then back to 'Very high', the problem seems to be fixed.