Solovay-random pairs of reals

Suppose $\kappa$ is the least ordinal such that for some $A \in 2^{\kappa} \cap M[x] \cap M[y]$, $A \notin M$. You already have that $\kappa = cf(\kappa) \geq \omega_1$. Let $\tau$ be a Random name for $A \in M[x]$. WLOG, assume that the empty condition forces $\tau \in 2^{\kappa} \setminus M$ and all initial segments are in $M$. In $M$, choose $\langle (p_i, a_i) : i < \kappa \rangle$ such that $p_i$ forces $a_i = \tau \upharpoonright i$. Since random forcing is $\kappa$-Knaster, there exists $X \in [\kappa]^{\kappa}$ such that $\{p_i : i \in X\}$ has pairwise compatible conditions so that $a = \bigcup \{a_i: i \in X\} \in 2^{\kappa}$. Using cccness, choose a condition $p$ that forces $|\{i \in X: p_i \in G\}| = \kappa$. But now $p$ forces $\tau \in M$: Contradiction.

This is a response to Kanoveri's comment. If $\{\kappa_n : n < \omega\}$ is strictly increasing with limit $\kappa$, then since $A \upharpoonright \kappa_n \in M$ (by minimality of $\kappa$), using ccc, we can find a countable $X \in M$ such that $M[x] \cap M[y]$ contains a new subset of $X$ which translates to a new real.