Galacticare

Edit: After obtaining this achievement, it turns out that you only need 7 out of 8 species to reach 4/5 reputation ranks. This includes Colossals, so the Xil aren't actually needed. I'm just keeping the guide up as it has the correct information, you can just start working on it earlier than chapter 8. I'd recommend starting the suggestions below as soon as you come across the Kouber Baly. As they were the last one for me to unlock due to their aversion to narrow corridors giving less influence.

This will absolutely be the last achievement that you'll unlock in this game. It's extremely time consuming.
So much so that I've got 5 stars in all hospitals including the sandbox levels, researched all pathogens to their max levels, unlocked every other achievement, since then I've had to leave my game running for 5 hours as this slowly ticks up, and I'll likely need to leave it running for another 7 hours to get this unlock. You also can't just leave the game running unattended as you seem to either get kicked for inactivity or the game crashes after 20-30 minutes of no button presses.

Just a few things I wish I knew that could've sped this up, to hopefully speed up your earning of this achievement:

Fortunately, you don't seem to need to actually max out all 5 reputation ranks for each species. My progress bar towards the achievement is jumping up when each species reaches 4/5 reputation ranks.

Any consultant abilities that increase reputation gains should absolutely be taken as soon as possible. Research progress and credits are the usual alternatives. Neither of which give even nearly as much of a headache as leveling reputation.

There's no point on actively working towards this achievement until chapter 8. As the biggest issue is raising the reputation of the Xil, who only become available in chapter 8. Everything else will reach 4/5 before the Xil.

From chapter 8 I would advise that every single doctor that you hire should have the diplomat special ability (the green thumbs up) as thats one of the only things in the game that boosts reputation gains. Their negative ability really doesn't matter as long as you're not hiring anyone with the red thumbs up. Psychopath, sociopath etc didn't seem to make much of a difference either. The only one I actually avoided was accident prone, which gives a 2% chance to kill their patient with each treatment. If they're killing them, then they're not getting the reputation gain.

Doctor specialisation in a specific treatment area doesn't matter either, as you can just use the cobalt suppository to reset their specialisation to fit whatever room you're putting them in. Just remember that after you've done this, to lock the doctor to only work in their assigned room. (The little padlock button up the top left of the doctors information panel)

You'll need to work towards it for the 5* in every hospital anyway, but fulfilling species desires also gives a slight reputation boost.

And that's it. Having the diplomat trait on every doctor (from chapter 8 onwards) and making sure the doctor with the correct training is working in each room, picking reputation gain abilities on your consultants, and fulfilling all species desires. They're the only things that boost reputation gains.

You'll still likely need to leave your game running for a few hours to unlock this once you've finished getting everything else. But hopefully this cuts it down to 3-4 hours rather than the 12 hours I'm having to sit through.