The Boolean prime ideal theorem does not imply the extension of almost disjoint families to MAD families

We establish that the statement “For every infinite set $X$, every almost disjoint family in $X$ can be extended to a maximal almost disjoint (MAD) family in $X$” is not provable in $\mathsf {ZF}$ + Boolean prime ideal theorem + Axiom of Countable Choice.

This settles an open problem from Tachtsis [On the existence of almost disjoint and MAD families without $\mathsf {AC}$, Bull. Polish Acad. Sci. Math. 67 (2019), 101–124].