To unlock the Shroomtender Sash, you need to complete the Syzygy achievement. Note that this achievement is also a requirement for the Traveler of Strange Roads achievement, which is needed to unlock the Shroomtender Epaulets.
This achievement takes place within the Apocrypha zone. You need to locate the four Aberrant Hushed within Apocrypha, and perform a ritual with each of them. You will find them praying in front of a strange altar. After completing all four, you will unlock a final ritual. Finishing the final ritual will unlock the achievement.
Below are the locations for all rituals.
Syzygy Achievement Ritual Locations
The first four rituals that you need to complete are marked in red on the map. You can complete these in any order.
After completing all four rituals, you then have to go to the Syzygial Rostrum, marked below in blue, to complete the final ritual and earn the achivement and Shroomtender Sash.