Here is the full list of all 10 Tangle Tower achievements. It takes around 6-8 hours to unlock all of the achievements on Windows.
Find your way into the secret room
Find three sets of similar research notes
Listen to Poppy's poems
Talk to the suspects about Tangle Tower
Talk to the suspects about your Case File
Find three statues of a certain amphibious creature
Find the Tangle Tower snowglobe
Look at the Family Tree and the drawing board
Begin investigating Flora's Tower