Here is the full list of all 73 The Red Solstice achievements.
You have finished single player in brutal hardcore mode
Win the game in Hardcore, Primary, Secondary missions with 4 marines, and finish all objectives. Defeat Vivian Parks, or win normal difficulty alone.
You have finished single player in hardcore mode
You have completed the industrial sector on soldier difficulty or higher.
You have completed the science sector on soldier difficulty or higher.
You have completed the Project Terra storyline succesfully.
You have mastered all 8 classes.
You have taken out hellspawn with SHIVA weapon.
You have completed Chasing echoes mission
You have completed Castel nuovo mission
You have completed Bonds which bond mission
You have completed V.L.A.K. transport mission
You have completed The fabian strategy mission
You have completed Smoke and mirrors mission
You have survived, and that's what it counts for!
You got on the field, you took them down in one round, 50 high threat kills.
You have contributed to salvation of Tharsis, whether for the good of it, or for the bad.
You have completed the Insurgents storyline succesfully.
You know your way around, you've been here for some time.
You have completed Stonewall mission
You have completed Balance of power mission
You have seen it, and you have killed it!
You have completed S.A.R.A. mission
You have completed Colony dark mission