Dominating reals: another low-level Q To be precise, let the dominating forcing $D$ consist of all pairs $(m,f)$, where $m<\omega$ and $f\in\omega^\omega$, with the order $(m,f)\le(m',f')$ ($(m',f')$ is stronger) iff $m\le m'$, $f(k)\le f'(k)$ for all $k$, and $f'|m=f|m$. 
Assume that $a\in\omega^\omega$ is dominating-generic over a CTM $M$ (that is, ($M\cap D$)-generic) and $b\in\omega^\omega$ is dominating-generic over $M[a]$ (that is, ($M[a]\cap D$)-generic). 
Q1. Is $b$ then dominating-generic over $M$, a submodel of $M[a]$?
A1. Yes. 
Q2. Is it true that $a\in M[b]$?
A2. No, and in fact $M[b]\cap M[a]=M$. 
Q3, motivated by A2. Is $a$ in any way generic over $M[b]$?
A3. IDK, and surprisingly this Q does not seem to be an easy one.
Q4. Clearly $a+b$ (termwise addition) is generic over $M[a]$. Is it true that $M[a+b]\cap M[b]=M$? Or weaker, $M[a+b]\cap M[b]\cap2^\omega\subseteq M$?
A4. IDK, ditto. 
 A: Regarding question 3, it is a general and nontrivial fact that if $M\subseteq M[G]$ is any forcing extension and $N$ is an intermediate transitive model of ZFC, with $M\subseteq N\subseteq M[G]$, then $N$ is a forcing extension of $M$ and $M[G]$ is a forcing extension of $N$. This is proved, for example, in Corollary 15.43 in Jech's book (see also Fact 11 in my article, Set-theoretic geology, where we give a proof).
In your case, you have the two-step forcing iteration $M\subseteq M[a][b]$, which has $M[b]$ as an intermediate model. So $a$ is definitely generic over $M[b]$, and the forcing is a quotient of the two-step forcing giving rise to $a*b$.  
A: Q4 is solved in the positive: $M[a+b]\cap M[b]\cap 2^\omega\subseteq M$. A point of dissatisfaction is that the most natural way to establish the result, that is, prove that $(a+b,b)$ is $(D\times D)$-generic (product-generic) over $M$, still does not go through. OOps - this fails because if $(a+b,b)$ is $(D\times D)$-generic then $a=(a+b)-b$ is Cohen-generic, contrary to the choice of $a$.
By the way, the result $M[a+b]\cap M[b]\cap 2^\omega\subseteq M$ is a key lemma in my proof that it is true in the dominating-generic extension that every countable OD set of reals consists of OD reals, arxived http://arxiv.org/abs/1609.01032.
