TWO DAYS LEFT
The flags to trigger the availability of the RECYCULTIST'S HQ hinge on completing one of the sidequests in FARAWAY TOWN. On the afternoon when there's TWO DAYS LEFT, head to the house two doors away from BASIL'S HOUSE. You'll find a woman standing outside. Talk to her and she'll complain of her son taking too long to purchase a FLOOR LAMP. Then, head to FIX-IT to find the STRANGE MAN, who comes to a revelation that the world is "trash" and he leaves it to you to purchase the FLOOR LAMP in his stead. Purchase the FLOOR LAMP and return to the woman to finish the sidequest.
If you pass by FARAWAY PARK afterwards, you'll notice a new individual with the RECYCULTISTS, who calls himself the RECYCLEPATH.
ONE DAY LEFT
With that sidequest complete, the RECYCULTIST'S HQ becomes available in the HANGOUT SPOT behind FARAWAY PARK in the morning. While you're in the dungeon you can engage in combat with the real-life party and can even tag your party leader around for new tag pictures. Act with care as your ability to restore HEART or JUICE is limited.
After defeating the RECYCLEPATH, he sees the error of his ways and realizes that his ideal "trashless world" can be made by improving his environment. Then SUNNY awakens in the HANGOUT SPOT, and the whole dungeon was apparently a long daydream. The player gets the Reduce, reuse, and recycle! achievement for this.
Later that afternoon, the RECYCLEPATH can be found in the corner of FARAWAY PARK, this time preaching a slightly different message for his trashless world.