Does tropicalization exist in the world of perfectoid spaces? Since it does for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any explicit references so far. 

For concreteness let $K$ be the completion of $\mathbb{Q}_p(p^{\frac{1}{p^\infty}})$, and $X$ a perfectoid space over $K$. Can one tropicalize $X$ and its tilt $X^\flat$? If so, how are the two tropicalizations related? Are they isomorphic (as rational polyhedral cone complexes)?

Further, what is the information tropicalization would retain in this setting? Is there a shadow left of the Galois action on $X$ and $X^\flat$?