Does tropicalization exist in the world of perfectoid spaces? Since it seems to for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any direct 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 the same (as rational polyhedral cone complexes or similar combinatorial objects)?

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