So a common method used to construct non-zero $omega$-REA *arithmetic* degrees with various properties is to build an $omega$-REA operator $J$ satisfying the constraints that (for all $X$)

$$tag{1} J(X’) equiv_T J(X) oplus X’$$

$$tag{2} J(X) >_T X$$

Inductively, 1 implies that $J(X^n) equiv_T J(emptyset) oplus X^n$. Thus, together, these constraints ensure that $J(emptyset)$ isn’t arithmetic (if $J(emptyset) leq_T 0^n$ then $J(0^n) equiv_T 0^n$).

My question is whether this is fully general, i.e., if $A$ is $omega$-REA is there some $omega$-REA operator $J$ satisfying 1 such that $A$ and $J(emptyset)$ have the same *arithmetic* degree? And if we can also assume 2 holds if $A$ isn’t arithmetic?

Basically, I’m hoping someone will let me know if I’m missing some obvious elementary result or known result before I spend any time trying a hard construction.