Mathias's question on MAD families under the Ramsey property

Let $[A]^\omega$ denote the set of all infinite subsets of $A$ for $A\subseteq\mathbb N$. A set $X\subseteq[\mathbb N]^\omega$ is Ramsey if there is some $B\in[\mathbb N]^\omega$ such that either $[B]^\omega\subseteq X$ or $X\cap[B]^\omega=\emptyset$. A MAD family is a maximal collection of infinite subsets of $\mathbb N$ whose distinct members have finite intersection. Solovay's model is a model in which every set is Lebesgue measurable, has the Baire property, and has the Ramsey property.

Mathias's question. In Solovay's model, are there infinite MAD families? More generally, is it a theorem of $\mathsf{ZF}+\mathsf{DC}_{\mathbb R}$ that there are no infinite MAD families assuming that all sets are Ramsey?

Mathias proved that no analytic MAD family exists and established consistency results concerning the non-existence of MAD families; the questions ask whether the strong regularity of all sets being Ramsey rules out infinite MAD families in Solovay's model or already under $\mathsf{ZF}+\mathsf{DC}_{\mathbb R}$.