TF2 supports hosting your own server without port forwarding, but this is a bit hidden. This guide covers how to do it:

Add -enablefakeip to your launch options. If you have done this already, you can skip this step.

Right click on Team Fortress 2 in the Steam library, click "Properties" then put
-enablefakeip
under "Launch Options".
https://i.imgur.com/cfUxk75.png
https://i.imgur.com/xMGL6aM.png

Launch Team Fortress 2. In the main menu, click "Find a Game" then "Create Server"
https://i.imgur.com/ZOpgmdA.png

Scroll to the bottom of the map list and select "(UGC) tf2ware_ultimate". If you don't see it, make sure you are subscribed to the map. You might need to restart TF2 to see it.

Then click on "Game" and set a password for the server. If you do not set a password, TF2Ware will generate one for you automatically.
https://i.imgur.com/jg1gPm9.png

Press start and the map should load!

After loading into the map, select Spectator and type !ware_ip in chat. This will give you a console command to join the server. Copy this (highlight it with your mouse and press CTRL + C), then send it to your friends by pasting with CTRL + V. Don't worry, this IP is anonymous and not your real IP.

NOTE: due to a TF2 bug, joining through the friends list won't work.

https://i.imgur.com/K9JKvyl.png

After your friends have joined by putting the command in console, join a team and the game will start. Type !ware_help if you want to see extra commands.

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If you or your friends don't know how to open the console, here is a quick guide:

Open the options on TF2's main menu.
https://i.imgur.com/go5urD3.png
Select "Advanced".
https://i.imgur.com/ujDPJGN.png
Tick "enable developer console"
https://i.imgur.com/DOp0gZ1.png

Then press the Tilde key ( ~ ) or Backtick key ( ` ) on your keyboard to open the console.