= Solution
The <diamond theorem in the constructible universe> gives $L\models\diamondsuit$, and $L$ satisfies <ZFC>. Apply the preceding construction inside $L$ with $S=\omega_1^L$. It produces a normal splitting <Suslin tree>.
For completeness, such a tree yields a <Suslin line>. Order its nodes lexicographically using the two successors at every split, treating a node itself as a position between its two successor subtrees. Each node is thus a cut point between a left and a right subtree. This gives a <dense linear order without endpoints>. Every nonempty interval contains a whole cone above some node: for comparable endpoints use the successor cone of the descendant endpoint directed toward the other endpoint; for incomparable endpoints use the right-successor cone of the lower endpoint. Disjoint intervals therefore supply pairwise incomparable cone roots, so the order has the <countable chain condition for a linear order>. A countable collection of nodes has bounded heights; a cone based above that bound contains none of them, so it is not an <order-dense subset>. Passing to the <Dedekind completion> using proper cuts, so that no endpoints are added, preserves density, the <countable chain condition for a linear order>, and nonseparability. For nonseparability, a countable dense set in the completion would give a countable dense set of original nodes by choosing one original node between each distinct pair of its points. This contradicts the preceding height-bound argument. The result is a <Suslin line>.
Thus the <Suslin hypothesis> fails in $L$. The <constructible universe theorem> is a theorem of <ZFC>, so this is a relative-consistency argument, without an additional assumption that a transitive model exists:
$$
\boxed{\operatorname{Con}(\mathrm{ZFC})\Longrightarrow
\operatorname{Con}(\mathrm{ZFC}+\neg\mathrm{SH}).}
$$
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