The Together we are strong badge is for "Performing 50 Co-op moves"
Co-op moves are actions that require 2 players interacting to complete it, such as: bashing doors, pulling levers and pushing carts.
There are a set number of these per trial so can't really be accelerated but as long as you're playing in co-op you will likely rack this up pretty quick.
Online matchmaking is available but of course results may vary.
Also two handy things to note:
1. The in-game badge menu tracks your progress.
2. Leaving a trial after gaining progress still counts towards it.
Video guide for reference: