  We study the percolation properties of geometrical clusters defined in the
overlap space of two statistically independent replicas of a square-lattice
Ising model that are simulated at the same temperature. In particular, we
consider two distinct types of clusters in the overlap, which we dub soft- and
hard-constraint clusters, and which are subsets of the regions of constant spin
overlap. By means of Monte Carlo simulations and a finite-size scaling analysis
we estimate the transition temperature as well as the set of critical exponents
characterizing the percolation transitions undergone by these two cluster
types. The results suggest that both soft- and hard-constraint clusters
percolate at the critical temperature of the Ising model and their critical
behavior is governed by the correlation-length exponent $\nu = 1$ found by
Onsager. At the same time, they exhibit non-standard and distinct sets of
exponents for the average cluster size and percolation strength.
