This is actually a question I am quite concern with at the moment. In its generality, let me give a slight re-hash of what Joel said, with perhaps a slight generalization.$\newcommand{\forces}{\Vdash}$

We say that $\Bbb P$ is $\kappa$-closed if every decreasing sequence of length $\kappa$ (or shorter) has a lower bound. We say that $\Bbb P$ is $\kappa$-Baire (or $\kappa$-distributive) if the intersection of $\kappa$ many dense open sets is dense (and open, trivially).

In $\sf ZF$ we can prove these two theorems:

**Theorem I.** If $\Bbb P$ is a $\kappa$-Baire forcing then $\Bbb P$ does not add new subsets of size $\leq\kappa$ to the universe.

*Proof.* Suppose that $p\forces\dot f\colon\check\kappa\to\check V$. By the definition of the forcing relation, for every $\alpha<\kappa$ we have a dense set $$D_\alpha=\{q\leq p\mid\exists x\in V, q\forces\dot f(\check\alpha)=\check x\}.$$

Let $D$ be the intersection of these dense open sets, then by the $\kappa$-Baire property $D$ is dense, and therefore non-empty. Let $q\in D$ then for every $\alpha<\kappa$ there is some $x\in V$ such that $q\forces\dot f(\check\alpha)=\check x$. However $q$ forces that $\dot f$ is a function, therefore this $x$ must be unique. If so the function $g(\alpha)=x$ for which $q\forces\dot f(\check\alpha)=\check x$ is such for which $q\forces\check g=\dot f$, as wanted. $\square$

**Theorem II.** $\sf DC_\kappa$ holds if and only if every $\kappa$-closed forcing is $\kappa$-Baire.

*Proof.* Suppose that $\sf DC_\kappa$ holds, the proof goes the same as in $\sf ZFC$. In the other direction, recall that $\sf DC_\kappa$ is equivalent to the following statement:

- Every tree $T$ of height $\kappa$, where every sequence of $<\kappa$ has a proper end extension has a branch of length $\kappa$.

Suppose now that $\sf DC_\kappa$ fails, then there is a tree $T$ of height $\kappa$ in which every sequence can be extended, but there is no branch of length $\kappa$. Define $P$ to be the reverse tree order on $T$. This forcing is trivially $\kappa$-closed, because there are no decreasing sequences of length $\kappa$. On the other hand, consider the dense sets $\{q\in T\mid\operatorname{lvl}_T(q)>\alpha\}$. Each is dense open, but their intersection is empty. $\square$

Finally, for real numbers. Or more specifically, sets of ordinals. If a forcing is $\kappa$-Baire then it doesn't add any new sets of size $\kappa$, but we are only interested in sets of ordinals.

I haven't managed to figure out the exact property, and working with $\kappa$-Baire forcings seems reasonable enough (and controllable enough).

To your subquestion, without verifying the details (I might do so tomorrow, but feel free to let me know if you do it yourself and find a mistake).

Consider the forcing which adds $\aleph_1$ Cohen generics, i.e. $p\in\Bbb P$ is a finite function from $\omega_1\times\omega$ to $2$, with the usual order. Now apply permutations (perhaps with a finite or countable support) of $\omega_1$ onto the left coordinate of the conditions.

Consider the names which are fixed by a countable subset of $\omega_1$. That is to say, there exists $\alpha<\omega_1$ such that whenever $\pi$ fixes pointwise $\alpha$ (as a set), it fixes the name.

I suspect that you can prove that all the real numbers of $V[G]$ enter this symmetric extension (perhaps by some argument that every real number is decided by some countable subset of the forcing, and therefore we can find it a support).

It is easy to show by the usual arguments, though, that there is no well-ordering of $\Bbb R$ in the symmetric extension.