The following solution was taken from PowerPyx's trophy guide - full credit to PowerPyx.
This trophy is earned in a way that requires making some specific choices during the game leading up to the point you earn it. It’s mutually exclusive of Ludwig was Right, and isn’t something you can set up with save backups so I would recommend aiming to get those two in separate full playthroughs.
For this trophy at the start of the game you need to choose to redirect power to Edgewater, then convince the deserters to return but not have Reed step down as leader. Through story progression you then need to ally with Sophia and the board by turning in Phineas, then once you have dealt with the radio broadcasts on Monarch Sophia will congratulate you on helping the town next time you see her. You don’t need to do anything else for this trophy past that, you are awarded it immediately before she asks you to head to the Hope.Edit:
Apparently you don't need to turn Phineas over to the Board, you can instead send the corrupted signal from Phineas's terminal, Sophia will believe you still did what she asked, and still unlock the achievement. - Thanks to Sasha Morning in the comments.Edit 2:
Fir3shad0w has provided an option for obtaining this achievement and "Ludwig was Right" in a single playthrough. Although I have not confirmed this:
I can confirm this can actually be set up alongside "Ludwig was Right" in the same playthrough. Go through the steps for this achievement, and once you've sent the signal (corrupted or not), save the game. Talking to Sophia will earn you this achievement (as long as the deserters returned and Reed is still in charge), then re-load the immediate save, go to Edgewater, and kill Reed. Head back to Sophia, and she'll give you the "Foundation" quest, which will earn you the other achievement.