Couldn't find a guide anywhere for this when I started trying to figure it out so I figured I'd make one. OpenMW and tes3mp run native on Linux so it seemed like running on the Steam Deck would be pretty straightforward to setup since it runs on a version of Arch. There's a few hurdles that'll slow you down so hopefully this helps.

First you need to go to Desktop mode and do the standard setup of Morrowind, OpenMw, and TES3mp:

1. Install Morrowind from Steam
2. Install the OpenMW flatpak from the Discover app
3. Download and setup the most recent GNU+Linux-x86 package from the TES3MP releases page
   1. Make sure to launch the setup wizard and point it to your main game file (mine was /home/deck/.local/share/Steam/steamapps/common/Morrowind/Data Files/Morrowind.esm)

This is where the hurdles come in. At this point you should be able to launch the server browser and select a server, but trying to connect will crash the app and not launch the game. You can see the output by using the terminal to launch the server browser. For me this resulted in an error about a missing .so library file. I've listed the commands that fixed it for me but if you get errors about other libraries you should install those in the same way.

1. Open the terminal
2. Setup deck to use pacman for library installs (source)
   1. Use passwd to create a password for the deck user
   2. Disable steamos readonly mode sudo steamos-readonly disable
   3. Initialize the pacman keyring sudo pacman-key --init
   4. Populate the pacman keyring with the default Arch Linux keys sudo pacman-key --populate archlinux
3. Install missing libraries sudo pacman -S --needed zvbi twolame wavpack libgme libopenmpt chromaprint
4. Try launching the server browser and connecting to a game. If it still isn't launching run tes3mp-browser from the terminal so you can watch the output. You may end up needing to install more/different libraries than I did.
   1. cd to your TES3mp dir (cd ~/Downloads/TES3MP for me)
   2. Run the browser with ./tes3mp-browser
   3. Install missing libraries with sudo pacman -S [libName]
5. Once the game is launching correctly, reenable steamos readonly mode sudo steamos-readonly enable

Now you should be able to actually connect to servers and get into the game. Setup the tes3mp-browser as a non-Steam game in Steam and now you can launch from game mode. Note that Steam Deck updates will override your changes to the readonly memory requiring you to run the steps above again.

After each SteamOS update the readonly memory will be reset and those packages you just installed will be wiped. As long as you've run the initial setup once, the fix is easy:

1. sudo steamos-readonly disable
2. sudo pacman -S --needed zvbi twolame wavpack libgme libopenmpt chromaprint
3. sudo steamos-readonly enable

edit: Updated library install command with u/Papa-Bonez 's suggestion

Original guide I migrated this from: https://www.reddit.com/r/tes3mp/comments/10rdcst/guide_for_running_tes3mp_on_a_steam_deck/