Assume $\sf GCH$.

Let $\kappa$ be a regular cardinal, we say that $\{A_\alpha\mid\alpha<\kappa^+\}\subseteq\mathcal P(\kappa)$ is an almost disjoint family, if whenever $\alpha\neq\beta$, $A_\alpha\cap A_\beta$ is bounded in $\kappa$.

A family is MAD if it is a maximal almost disjoint family.

Given an almost disjoint family, we say that it is a *broken family* if whenever $A\subseteq\kappa^+$ is bounded, there is a refinement $B_\alpha\subseteq A_\alpha$ (and $A_\alpha\setminus B_\alpha$ bounded) for $\alpha\in A$, such that $\{B_\alpha\mid\alpha\in A\}$ are a pairwise disjoint family with the property that $A_\xi\subseteq^*\bigcup_{\alpha\in A}B_\alpha$ if and only if $\xi\in A$.

We can produce broken families from towers, as the pointwise difference. But those are not maximal.

- Is it provable, or at least consistent (with $\sf GCH$), that there are almost disjoint families which are
*not*broken? - How about MAD families? Can we prove that every MAD family is not broken, or at least consistently obtain broken MAD families?

(If the general case is a bit too hard, I'd be interested in the case for $\kappa=\omega$ as a particular case.)