Tighten Up The Graphics
This is the same basic idea as the Halo: CE achievement seeing in both remastered and classic.
Note: This achievement seems to be a bit buggy, so when going for this one, doing any achievements make sure to stay in each mode for at least 5 minutes each, I went through the campaign constantly going in and out, finished with 88% progress, TA progress bar hasn't budged. If you find the answer please let me know via comment or private message, as soon as someone finds the answer or I unlock it myself I will update the solution. I did every mission doing this in Co-op so if you go for this, try doing solo, if it doesn't unlock, try Co-op.
button just go in and out on each level before finishing each mission and when finishing the campaign you will unlock this achievement.
When going for this I go in and out of classic and remastered when I get to the skulls. In this I love to play on remastered most of the time, but when it comes to those skulls, just go back and forth. Or what a friend and I liked to do is compare the cut scenes the difference is insane.
If you die before making to a checkpoint make sure to do it again (if you did it and it didn't save).
There is no exact way to know which missions you did both ways so you will have to try and guess which one you missed and mark them off on paper or Notpad on your computer/laptop so you don't keep trying the same missions. (When you mark them off make sure you were at least 5-10 minutes in classic and Remastered mode.
In Halo CE theres a method to do every level instead of finishing all the missions until you obtain the achievement, instead swap the mode, get to a checkpoint without dying, try the next mission.
Note: This only works if you finished the level from beginning to end already. So give it a try and comment if it works for you
Good luck and have fun