Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic filters $\{ G_r : r \in \mathbb{R} \}$?

If CH holds, then the answer is yes. Recursively on the countable ordinals, we construct models $M_\alpha = M[G_\alpha]$, where $G_\alpha \subseteq \mathbb{P}$ meets all dense sets in $\bigcup_{\beta<\alpha} M_\beta$.

We can also construct $2^\omega$ many distinct generic extensions. We can build (externally) a tree $T \subseteq \mathbb{P}$, where $T = \{ p_s : s \in 2^{<\omega} \}$. Make $T$ such that: **(a)** If $|s| = n$, then $p_s \in D_n$, where $\{ D_n : n \in \omega \}$ lists the dense subsets of $M$. **(b)** For $s$ a prefix of $t$, $p_s \geq p_t$. **(c)** For all $s$, $p_{s0} \perp p_{s1}$. Every real $r$ determines a distinct branch through $T$ and an associated $\mathbb{P}$-generic filter $G_r$. So we have $2^\omega$ many distinct filters, and since the models are countable, a pigeonhole argument shows that we must have $2^\omega$ distinct models. But this argument does not show that the filters are mutually generic.