I want to understand an apparent discrepancy in the literature regarding forcing and boolean-valued models of set theory.
In Kunen's book on forcing (1980 edition), the following appears as Exercise B18 in Chapter VII:
In Approach (2c) of $\S$9, show that $\mathbf{V}^\mathcal{B}/G$ is well founded iff $G$ is countably complete.
Here, "Approach 2(c) of $\S$9" refers to his treatment of boolean-valued models over $\mathbf{V}$, $\mathcal{B}$ a complete boolean algebra, and $G$ any ultrafilter on $\mathcal{B}$.
However, in Bell's book on boolean-valued models and independence proofs (1985 edition), you find the following as Corollary 4.7:
If $U$ is $M$-generic, then $\in_U$ is a well-founded relation.
Here, $M$ is a transitive (set) model of ZFC, $\in_U$ the induced membership relation on $M^\mathcal{B}/U$, where $\mathcal{B}$ is a complete boolean algebra in $M$, and $U$ any ultrafilter on $\mathcal{B}$.
The discrepency is that if $\mathcal{B}$ is, say, the boolean completion in $M$ of the usual Cohen partial order $\mathrm{Fn}(\omega,2)$, and $G$ is $M$-generic for $\mathcal{B}$, then $G$ is not countably complete, even though the latter result says that $M^\mathcal{B}/G$ is well-founded.
What am I missing here? And is there a coherent way of describing the relationship between genericity, countable completeness, and well-foundedness of resulting model?