Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable: $$A,B\in \mathcal A, A\neq B \Rightarrow A\cap B \in \mathcal I. $$

Let us observe that maximal ideals have this property as only the empty family or the singletons meet this requirement.

Are there further examples of ideals with this property?

(As observed by Leonetti every such ideal must be non-meagre when regarded as a subset of the Cantor set.)