Just to reconfirm it, - it works if you do it with someone who does not have the quest.
After missing it for 2 weeks, I finally managed to get it.
There were two of us, but only I had the quest.
False History was my 3rd quest in Stage 1 and Landing Party was 1st in Stage 2.
Also I completed bonus objectives in both quests just in case.
The achievement ticked right after finishing Landing Party.
So if you have the bug, like I had, this workaround does help.
PS: If you need a "questless assistant" on The Red Eclipse, I can help - Avionius or Tholn in game.