Full-splitting miller trees and infinitely often equal reals

We investigate two closely related partial orders of trees on ωω: the full-splitting Miller trees and the infinitely often equal trees, as well as their corresponding σ-ideals. The former notion was considered by Newel-ski and Ros lanowski while the latter involves a correction of a result of Spinas. We consider some Marczewski-style regularity properties based on these trees, which turn out to be closely related to the property of Baire, and look at the dichotomies of Newelski-Ros lanowski and Spinas for higher projective pointclasses. We also provide some insight concern-ing a question of Fremlin whether one can add an infinitely often equal real without adding a Cohen real