Here we are considering subsets $\mathcal{F}$ of $2^\omega$, which are in correspondence with families of subsets of $\omega$ (sets of "reals"). Such a family is Borel if it is a Borel subset of $2^\omega$ under the usual topology.

Such a family is almost disjoint if, for every pair $X\not=Y$ from $\mathcal{F}$, the symmetric difference of $X$ and $Y$ is infinite.

Countable almost disjoint families can be constructed fairly trivially. Uncountable almost disjoint families exist, and are a standard object of study in some branches of set theory. However, the constructions I've seen do not result in a Borel set. Can this be done? Can $\mathcal{F}\subset 2^\omega$ be an uncountable Borel set which is an almost disjoint family?