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Easily the best meta in this raid, such a fun twist on an already great fight and much like the rest of the fight, the achieve centers around correct usage of the Dimensional Tears (which will be called portals for the rest of this post).

Before pulling the boss, you need to collect the three bits of loose anima just outside the boss' room. You can see them in this screenshot or on this map. It doesn't matter who picks them up, and the person who picks them up does not have to be the one to handle them during the fight.

Clicking the anima will give you a buff, Maldraxxus Anima, Ardenweald Anima or Maw Anima. Depending on the phase the boss is in, the Anima buffs will be removed and it will be teleported to the middle of the room. When the Anima is stripped it will also create a zone on the ground that whoever grabs the Anima from the middle of the room must run it to. However, when you pick up the Anima in the middle of the room you are permanently stunned until you deposit it. This means that after it gets picked up, the next set of portals need to drop themselves on the player who clicked the Anima and the deposit point respectively, which will teleport the Anima holder there, and return it to the display case.

The Maldraxxus Anima is taken 10-15 seconds after the boss is pulled. You can see the deposit point in this screenshot.
The Ardenweald Anima is taken 10-15 seconds after the boss enters P2 at 70%. You can see the deposit point in this screenshot.
The Maw Anima is taken 10-15 seconds after the boss enters P3 at 40%. We didn't get a screenshot of the deposit spot active but it's where purple marker is in this screenshot.

Biggest thing to remember is that with using the portals to move your Anima holder you will not have them to deal with Ghosts (P1), Seeds (P2) or Annihilate (P3). Ghosts you can just run from and Annihilate you can outrun with a Stampeding Roar or Windrush Totem on normal, but seeds will need to be picked up by mobile classes and moved across the room quickly.

Such a fun achievement. We had so much fun as we didn't understand how it worked when we got here. One of the best raid/dungeon metas I can remember doing.

FrostyShot#1667 if you have any questions about the achievements! (US Servers)

Awesome tip, discovered by my guild's Mage (shoutout to Smirdex-Ravencrest):

Pop [Alter Time] while stood on the deposit position (the circle that spawns in front of the relics as shown in the other helpful comments here) and then run over and click the anima orb in the middle of the room. You will be stunned as normal, but when Alter Time expires you will be repositioned on top of the deposit and the achievement will proceed. This allows you to do the achievement without any juggling of portals, as the Mage can handle it 100% solo.

This achievement is absolutely soloable contrary to what other comments say.
TL;DR
Use an ability like heroic leap to grab the orb mid air as you leap into the circle.

Video explanation on how to do it easy:

https://youtu.be/ADN5nUS8By0