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 $\mathcal{F}$, $X\cap Y$ is finite.

*Note:* the question originally only required that the symmetric difference between $X$ and $Y$ be infinite, which is substantially weaker.

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?