Join of recursively enumerable set and its complement
The operation you described in actually the join operator in the semilattice of Turing degrees, defined as $A\sqcup B=\{0x | x\in A\}\cup \{1x | x\in B\}$. It is not hard to show that $A,B\le_T A\sqcup B$, thus if $A\in RE\setminus R$ then $\overline{A}\notin RE$ and thus $A\sqcup\overline{A}\notin RE$.