This is more of a theory than a definitive guide but since so many people have been earning this achievement in such a short time I believe this is the case. If you join someone’s game and visit their base I believe the achievement should unlock.
Now as of right now I cannot confirm nor deny this as:
A. None of my friends actually play No Man’s Sky and they never will
B. I believe the “boosting” platform for this game is going to be nonexistent
So unless someone can comment if they’ve gotten the achievement this way or eventually I can attempt to do this I will edit this solution to confirm or deny if this works
If you don’t know and want to be the one to help me, send me a message over Xbox Live and we can choose a time to try it out, until then if someone comments saying it does work then I’ll update it
EDIT: I CAN CONFIRM THIS METHOD DOES WORK BUT WITH A FEW EXTRA STEPS
Thanks to Acejunky for helping me understand this and get this done.
Ok so here's exactly what you need to do for this achievement. For starters this achievement will only unlock in a solo game so you can't just join your friend and go to their base. Here's a step-by-step process:
1. Join your friends game (must be in same difficulty and progression
2. Go to their base (tag it if you need to)
3. It must be a solo game, so have your friend disconnect real quick leaving you there by yourself at their base
4. Interact with their terminal and Reunion should unlock
If for some reason it still refuses to unlock head to the space station and interact with the portal to make sure you have the system known. Then head back to your base, save and quit. Then reload and warp back to the system where their base is and then head down to the base. Upon interacting with the terminal it should unlock if for some odd reason it didn't unlock with the method above. The step-by-step process is how it unlocked for me making this achievement pretty trivial as back on PS4 it required either the Galactic Hub or using the planetary portal system with the glyphs from Traveler graves.
As of 7-28-18 with the latest patch this still works so as patches roll out please feel free to inform me if it no longer works.
People have been a little confused as to why you have to be in a solo game and I figure I'd put this here as my comment could potentially become buried. The reason it needs to be a solo game is because when you're with your friend and interact with their computer it allows you to build and add on as it registers it as your base as well. Doing this allows you to edit the base and use it to your free will. Only in a solo game does the game fully register it as someone else's base