= Solution
Choose a root $o\in T$ and finite sets $F_n\subset T$ whose union is dense, arranging that $F_n$ is a $2^{-n}$-net and $F_n\subset F_{n+1}$. Let $T_n$ be the finite subtree spanned by $o$ and $F_n$. A depth-first contour traversal of $T_n$, recording distance from $o$, gives a continuous excursion $g_n$ whose <real tree encoded by an excursion> is $T_n$.
The traversals may be chosen compatibly: when passing from $T_n$ to $T_{n+1}$, insert the new branch traversals into small time intervals at their attachment points. Since every new component has height at most $2^{-n+1}$, choose the time changes so that
$$
\lVert g_{n+1}-g_n\rVert_\infty\leq 2^{-n+2}.
$$
After harmlessly taking a faster sequence of nets, these errors are <summable>. Hence $(g_n)$ is <uniformly Cauchy> and converges uniformly to a continuous $g:[0,1]\to\mathbb R_+$ with $g(0)=g(1)=0$.
The net property gives $d_{GH}(T_n,T)\to0$. By the stated continuity of excursion coding, $d_{GH}(T_{g_n},T_g)\to0$. Since $T_{g_n}$ is <isometric> to $T_n$, uniqueness of limits in the <Gromov-Hausdorff distance> implies that $T_g$ is <isometric> to $T$. This proves the <excursion coding theorem for compact real trees>.
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