A set $A \subseteq \mathbb{N}$ is recursively enumerable if there exists a $\mu$-recursive function which enumerates it. Two sets $A, B \subseteq \mathbb{N}$ are recursively inseparable if there does not exists a set $C \subseteq \mathbb{N}$ such that $A \subseteq C$ and $B \cap C = \varnothing$. That is to say, any superset of $A$ will necessarily also capture some of $B$ as is illustrated in the following figure.
How do I actually prove that there are recursively enumerable sets $A \subseteq \mathbb{N}$ and $B \subseteq \mathbb{N}$ which are recursively inseparable? I am currently working on a paper on non-standard models of Peano arithmetic and I am finding myself having to prove this as a lemma. In the proof, I want to avoid partial functions and Turing machines if possible, because I don't use Turing machines anywhere else and don't want to add an additional 3 pages to my paper discussing Turing machines.