UPDATE APR 2022:
Game is now playable under Linux, and any further issues or fixes needed should not be EAC related. Please see https://www.protondb.com/app/1063730 for more info and possible fixes for your specific issues.
UPDATE 30 MAR 2022:
The .so file for Linux EAC was added to the main game with today’s patch! 
THE FOLLOWING INSTRUCTIONS ARE AT YOUR OWN RISK
There has been no OFFICIAL announcement for Linux EAC support. It is possible that you may be banned from the game even though this doesnt break any rules or modify any game files.
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First, install Proton EasyAntiCheat Runtime in Steam (open your Library tab in Steam and search for EasyAntiCheat, install it just like you would for any Steam game)
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Next, you MUST be using Proton version 7.0 or newer (right click game > Properties > Compatibility > click the checkbox next to “Force the use of a specific tool” > set Proton version)
NOTE: Currently the game does not start without adding the following command to the game Launch Options (right click New World > Properties > Paste into Launch Options box)
bash -c 'exec "${@/NewWorldLauncher.exe/bin64\/NewWorld.exe}"' -- %command%
(Launch Option credit to Rezzafr33)
NOTE: This launch option opens the New World Launcher directly and the game then loads the Proton EasyAntiCheat runtime. THIS DOES NOT DISABLE EAC OR BYPASS THE NORMAL LOADING OF EAC BY THE GAME, IT JUST STARTS THE GAME LAUNCHER DIRECTLY
Note for players running on gaming laptops: if you experience poor performance your game might be using your CPU graphics instead of your dedicated GPU. You will need to add the following launch option BEFORE the one above:
DRI_PRIME=1
UPDATE 25 MAR 2022: Some people have gotten the Public Test Realm working on Linux with Proton. The fact that AGS has finally included the required easyanticheat_x64.so file with the PTR so this can be made to work is a great sign that they have finally taken our requests for Linux compatibility seriously. Hopefully we will be seeing official support soon!
UPDATE 23 JAN 2022: Update to EAC to allow simple Proton support according to Valve Software.
Source: Steamworks Development - Steam Deck Anti-Cheat Update - Steam News
Source2: Steam Deck and Proton (Steamworks Documentation)
UPDATE 9 JAN 2022: Current EAC version that supports Proton/Wine is NOT compatible with the EAC version that New World currently uses. The ‘just a click’ version that Epic Games advertised is for games running Epic Online Services ONLY and New World would need significant changes to be compatible with this version. Until Epic Games releases a version that is not reliant on Epic Online Services it is extremely unlikely that we will get the ability to play on Linux. Source: GamingOnLinux.com
With the launch of SteamDeck looming ever closer the interest in Linux and specifically gaming on Linux has grown at an amazing rate. Now that EAC has announced the availability of official support for Linux thanks to CodeWeavers and Valves effort, i would like to request that AGS add the appropriate libraries to the game so that it can be launched and played under Proton.
Currently the game will launch and works beautifully all the way through Character Creation with no noticeable issues under Proton 6.3-6 with no tweaks needed, but gets the error ‘EAC Initalization failure’ upon getting through queue and attempting to connect to a server.
EDIT: Just as a note, WE ARE NOT ASKING FOR A LINUX VERSION OF NEW WORLD. We just want the Anticheat to be enabled for Linux users. We fully understand that NW was not intended for use on Linux or other non-Windows based operating systems, but by using Steam’s Proton compatibility layer the game is seemingly playable with rumors of people being able to play as far back as the Preview event through Proton but unable to do so after the implementation of Easy Anticheat during the Beta testing phase.
