I just finished soloing Duality for the first time and did not get the triumph. I did not do it in one sitting, nor did I do it flawlessly, but I was completely alone during the entire dungeon.
Linked is my destiny tracker information and you can see in the PVE tab the duality instances that I was in on Aug 05 and Aug 06 had only me in the fireteam.
I have left the shattered throne before doing it solo to come back and finish it and received the triumph for that. So I am not sure why it did not function the same way for duality.
I would assume it is a bug with leaving and returning to dungeons that might cause this, but I am really frustrated at the fact that I completed this dungeon solo and did not get the associated triumph. After all of the other random glitches and bugs that occur in the dungeon that cause wipes undoing potentially an hour of an attempt I am amazed that I would run into an issue with the triumph not recognizing or tracking the triumph at all. Not getting the achievement of completing the dungeon solo just makes me think I have wasted my time on absolutely nothing.
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Your game history shows that it was a checkpoint and not a fresh single instance.