It is consistent with ${\sf ZF}$ that the reals are the countable union of countable sets. Since any countable set is Borel, it follows that in any such pathological universe, let's call it $W$, every set is Borel; this has come out here before.

Since there is a countable basis for the Borel sets, it follows that in $W$ there is a countable family of sets of reals, such that the $\sigma$-algebra they generate is all of ${\mathcal P}({\mathbb R})$. I would like to see whether we can improve this slightly:

Suppose ${\mathbb R}=\bigcup_n A_n$, where each $A_n$ is countable. Can we further assert that, in addition, the $\sigma$-algebra the $A_n$ generate is ${\mathcal P}({\mathbb R})$? More precisely, can we modify the $A_n$ to a new family $A_n'$ of countable sets with union ${\mathbb R}$ and this generating property?

Asaf's answer shows I'm being (somewhat) sloppy in this formulation. So, does it make sense to ask for some *intrinsic* construction? (Something other than: We start with such and such countable family that generates the Borel sets, we code it into bits of the $A_n$, and then we are done. I confess I do not see immediately how to formalize "intrinsic", perhaps it makes no sense.)