Consider a ctm $\mathfrak{M}$ of $ZF+AD^+$. Is it possible to force over $\mathfrak{M}$ to get a model of ZFC which satisfies further the following:


1. Every projectively definable family of sets of reals has an $OD\_a$ member;
2. Every $OD\_a$ set of reals has the property of Baire;
Where $a$ is any real parameter.


Here statement 1 is as in my previous question (Projectively definable family of sets of reals).


PS: I am asking this question because there is a paper by Steel & Van Wesep "Two consequences of determinacy consistent with choice" which uses a similar technique.



Forcing over models of determinacy
$\newcommand\R{\mathbb{R}}\newcommand\OD{\text{OD}}$


The answer is no. 


First, notice that the family consisting of all subsets $R\subset\R^2$ that are a well-ordering of the reals is projectively definable in your sense, since we can say that $R$ is a well-ordering by quantifying only over countable objects: $R$ is reflexive, transitive, anti-symmetric and linear, every real is mentioned, and there is no infinite descending sequence (and we can code such $R$ canonically with a subset of $\R$, to avoid the need for $\R^2$ here). Thus, if statement 1 holds, we get an $\OD\_a$ well-ordering of $\R$ for some real parameter $a$. But from any such well-ordering, we can define a set of reals, such as the Vitali set, that does not have the property of Baire, thereby violating property 2. 


