How similar are the c.e. degrees and the CEA(Cohen) degrees? Given reals $A,B,X$, let $A\le_{T/X}B$ iff $A\oplus X\le_TB\oplus X$. For each real $X$ we can define a version of the c.e. degrees over $X$: we look at the preorder on $X$-c.e. reals given by $\le_{T/X}$ (or, if preferred, take the corresponding partial order). Call this $\mathcal{R}_X$; equivalently, $\mathcal{R}_X$ could be given by the reals which are CEA $X$.
My question is the following:

Suppose $G$ is "sufficiently" Cohen generic. Is $\mathcal{R}_G\cong\mathcal{R}$?

Here $\mathcal{R}$ is the usual c.e. degrees. Incidentally, it's not hard to show that if $G,H$ are "sufficiently" Cohen generic then $\mathcal{R}_G\cong\mathcal{R}_H$. This is false.
I vaguely recall a result (due to Shore?) that the answer is no, but I can't track it down.
If the answer is no, I'm curious how similar they are nonetheless. For example, do $\mathcal{R}_G$ and $\mathcal{R}$ have the same $\Pi_3$ theories?
 A: I am reading ≅ as meaning "elementarily equivalent."
There are several questions here.  I believe that some of them are accessible by known methods.  The first is that if $X$ is not (low-level)arithmetically definable, then $\mathcal{R}_X$ is not elementarily equivalent to $\mathcal{R}$.  The proof should go by relativising the methods of
Nies, André ;  Shore, Richard A. ;  Slaman, Theodore A.  Interpretability and definability in the recursively enumerable degrees.
Proc. London Math. Soc. (3)  77  (1998),  no. 2, 241--291.
They show that there is a way to interpret the structure of first order arithmetic in the theory of $\mathcal{R}.$  Relativizing, the method should show that there is an interpretation of the structure of first order arithmetic with a predicate for $X$ in $\mathcal{R}_X$.  The elementary difference with $\mathcal{R}$ would be of the form, there is an interpretation of the structure of first order arithmetic with a unary predicate $U$ such that $U$ is not $\Sigma_k$, where $k$ is sufficiently large that it cannot be so represented in $\mathcal{R}$.  It should suffice that $k=10$.
You might look at Shore's paper,
Shore, Richard A.  Degree structures: local and global investigations.
Bull. Symbolic Logic  12  (2006),  no. 3, 369--389,
for a survey of known results and methods.
I agree that it would be interesting to know the exact complexity of the optimal elementary difference.  That question also touches on the old and open problem of deciding the two-quantifier theory of $\mathcal{R}$.  It would be very interesting if working above $X$ a generic, or a random, or anything at, made it possible to decide the two-quantifier theory of $\mathcal{R}_X$.
