= Solution
Let
$$
d=\bigcup\{s:(s,f)\in G\}.
$$
For each <natural number> $m$, conditions whose stem has length at least $m$ form a <dense subset of a forcing order>, so the <generic filter> meets all of them and $d\in\omega^\omega$.
Fix $h\in\omega^\omega\cap M$. The set
$$
D_h=\{(s,f):f(n)>h(n)\text{ for every }n\}
$$
is dense: from $(s,f)$ replace $f$ by $n\mapsto\max\{f(n),h(n)+1\}$. Choose $(s,f)\in G\cap D_h$. Every stronger condition must put each newly added stem value $d(n)$ above $f(n)$, so
$$
d(n)>h(n)
$$
for every $n\geq|s|$. Thus $d$ is a <dominating real> over $M$.
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