  We present a random-interface representation of the three-dimensional (3D)
Ising model based on thermal fluctuations of a uniquely defined geometric spin
cluster in the 3D model and its 2D cross section. Extensive simulations have
been carried out to measure the global interfacial width as a function of
temperature for different lattice sizes which is shown to signal the
criticality of the model at $T_c$ by forming a size-independent cusp in 3D,
along with an emergent super-roughening at its 2D cross section. We find that
the super-rough state is accompanied by an intrinsic anomalous scaling behavior
in the local properties characterized by a set of geometric exponents which are
the same as those for a pure 2D Ising model.
