A well-known result of Walsh states that if $\mathcal{T}^\ast$ is an ideal triangulation of an atoroidal, acylindrical, irreducible, compact 3-manifold with torus boundary components and $\mathcal{T}^\ast$ has essential edges, then every properly embedded, two-sided, incompressible surface $S$ is isotopic to a spun-normal surface in $\mathcal{T}^\ast$ unless $S$ is isotopic to a fiber or virtual fiber. For a given manifold $M$ that fibers over $S^1$, it was previously unknown whether there exists an ideal triangulation in which the fiber appears as a spun-normal surface. We prove that such a triangulation exists and give an algorithm to construct the ideal triangulation provided $M$ has a single boundary component.