This is probably a silly question. Call a set of reals $X$ a constructibility ideal (in analogy with a Turing ideal) if $X$ is closed under effective join $r\oplus s: n\mapsto 2^{r(n)}3^{s(n)}$ and relative constructibility. Clearly, if $X\subseteq \mathbb{R}$ is such that $L(X)\cap\mathbb{R}=X$, then $X$ is a constructibility ideal. My question is: does the converse hold?
I don't see that it does. For example, let $X_n$ be the set of reals which are (lightface) $\Pi^1_n$, and $X$ the set of (lightface) projective reals $X=\bigcup X_n$ - all computed in $V$, of course. Then the projective theory of $\omega$ will be in $L(X)\setminus X$, but if $V$ has enough large cardinals then I think $X$ is a constructibility ideal.
If not, what do we call a set of reals $X$ such that $L(X)\cap\mathbb{R}=X$?