Broken families 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.


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*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.)
 A: To answer your Question 1, under GCH there is always an almost disjoint family which is not broken. To construct such a family, begin with a partition $\{A_\alpha \mid \alpha < \kappa\}$ of $\kappa$ into pairwise disjoint, unbounded subsets. Enumerate each $A_\alpha$ in increasing order as $\{\eta^\alpha_\beta \mid \beta < \kappa\}$. By GCH, fix a sequence of functions $\vec{f} = \langle f_\gamma \mid \gamma < \kappa^+ \rangle$ that is increasing and cofinal in ${^\kappa}\kappa$ modulo the bounded ideal. Now, for $\gamma \in [\kappa, \kappa^+)$, let $A_\gamma = \{\eta^\alpha_{f_\gamma(\alpha)} \mid \alpha < \kappa\}$. Since $\vec{f}$ is increasing modulo the bounded ideal, the family $\{A_\alpha \mid \alpha < \kappa^+\}$ is almost disjoint. Suppose for sake of contradiction that it is broken. Then there is a function $g:\kappa \rightarrow \kappa$ such that, setting $B_\alpha = \{\eta^\alpha_\beta \mid g(\alpha) \leq \beta < \kappa\}$, we have that $\{B_\alpha \mid \alpha < \kappa\}$ witnesses brokenness for the set $A = \kappa$. But then, if $\gamma \in [\kappa, \kappa^+)$ is such that $g <^* f_\gamma$, we have $A_\gamma \subseteq^* \bigcup_{\alpha < \kappa} B_\alpha$. 
This seems like sort of a cheap example, though, since, as far as I can tell, the family might be made broken simply by removing the first $\kappa$ elements. Perhaps a more involved version of this construction could produce a more robust failure of brokenness.
