The Third Spire
From Destiny 2 Wiki
The Third Spire is the social space for Trials of the Nine.
Unlock
The Third Spire unlocks once your get your first win in Trials of the Nine.
Vendors
There are platforms, each with it's The Emissary. Ground level requires only 1 win, first platform requires 3 wins, second platform requires 5, third requires 7 wins and the Flawless will unlock inside of the spire. See Trials of the Nine#Rewards Per Win