  On the product elliptic threefold $X = C \times S$ where $C$ is an elliptic
curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit
tilt stability, which satisfies the Harder-Narasimhan property. We show that
under the Fourier-Mukai transform $\Phi$ on $D^b(X)$ induced by the classical
Fourier-Mukai transform on $D^b(C)$, a slope stable torsion-free sheaf
satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is
taken to a limit tilt stable object. We also show that a limit tilt semistable
object on $X$ is taken by $\Phi$ to a slope semistable sheaf, up to
modification by the transform of a codimension 2 sheaf.
