  We show that if $M$ is a countable transitive model of ZF and if $a,b$ are
reals not in $M$, then there is a $G$ generic over $M$ such that $b \in
L[a,G]$. We then present several applications such as the following: if $J$ is
any countable transitive model of ZFC and $M \not\subseteq J$ is another
countable transitive model of ZFC of the same ordinal height $\alpha$, then
there is a forcing extension $N$ of $J$ such that $M \cup N$ is not included in
any transitive model of ZFC of height $\alpha$. Also, assuming $0^\#$ exists,
letting $S$ be the set of reals generic over $L$, although $S$ is disjoint from
the Turing cone above $0^\#$, we have that for any non-constructible real $a$,
$\{ a \oplus s : s \in S \}$ is cofinal in the Turing degrees.
