In order to get the "...Bearded Travelers" trophy in Burly Men at Sea, you need to reach an ending in which the Burly Men are in the water. There are several routes which you will have to play through in turn to get other trophies anyway. Here's one example.
(Remember to make cross saves before trophy related decisions to drastically speed up your play throughs).
Here's the steps from when you enter the Whale, which is an unavoidable incident at the very beginning of the game:
-- Talk to the girls by the fire
-- Go all the way left until you see three barrels, interact with them
-- Interact with them again after the dialogue
-- Head right and go through the door behind your ship
-- Scroll right, get through the dialogue
-- Interact with the uvula, get through the dialogue
At this point you'll get:
-- Swim right, dialogue
-- Scroll right, dialogue
-- Move the cursor to the blonde Burly (Hasty Beard) and hold X until he falls back to last place (you don't need to follow him with the cursor once you are holding X)
At this point you'll get:
-- Head right
-- Interact with the Blue Orb
-- Go right and right again
-- Hold L2 (L on the Vita) to go in the left tunnel
-- Go right
The trophy should unlock around this point, along with: