= Solution
Let $T$ consist of all finite nondecreasing paths
$$
(s_0,s_1,\ldots,s_k),qquad s_0\leq_Ss_1\leq_S\cdots\leq_Ss_k,
$$
ordered by initial-segment extension. The one-point path $(s_0)$ is least, and the predecessors of any path are its initial segments, hence linearly ordered. Force an atom at a path exactly when it is forced at the path's endpoint.
The endpoint map $e:T\to S$ is monotone and has the back property: if $e(t)\leq_Ss$, append $s$ to $t$. Induction on propositions therefore gives
$$
t\Vdash_T\psi\iff e(t)\Vdash_S\psi.
$$
In particular the roots force exactly the same propositions. This is the <unravelling of a Kripke model>.
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