Separability of languages in RE
Hint: Such sets $A,B$ (sans the requirement of recursive enumerability) are called recursively inseparable. According to Wikipedia "it is possible for $A$ and $B$ to be recursively inseparable, disjoint, and recursively enumerable", and Wikipedia might have relevant examples.  (Even if not, now you know what term to look for in the vast sea of internet.)

Extra hint: Wikipedia does indeed provide two such examples:


$A$ is the set of encodings of TMs whose output is $0$, $B$ is the set of encodings of TMs whose output is $1$.
$A$ is the set of provable tautologies, $B$ is the set of refutable contradictions.