  The two-dimensional (zero magnetic field) Ising model is known to undergo a
second order para-ferromagnetic phase transition, which is accompanied by a
correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In
this paper we uncover that there exists also a second temperature
$T_{\text{eb}}<T_c$ at which the elastic backbone of FK clusters undergoes a
second order phase transition to a dense phase. The corresponding universality
class, which is characterized by determining various percolation exponents, is
shown to be completely different from directed percolation, proposing a new
anisotropic universality class with $\beta=0.54\pm 0.02$, $\nu_{||}=1.86\pm
0.01$, $\nu_{\perp}=1.21\pm 0.04$ and $d_f=1.53\pm 0.03$. All tested
hyper-scaling relations are shown to be valid.
