XOR-Ising Loops and the Gaussian Free Field

We find by simulation that the interfaces in the exclusive-or (XOR) of two independent 2D Ising spin configurations at the critical temperature form an ensemble of loops that have the same distribution as the contour lines of the Gaussian free field, but with the heights of the contours spaced √2 times as far apart as they are for the conformal loop ensemble CLE₄ or the double dimer model on the square lattice. For domains with boundary, various natural boundary conditions for the two Ising models correspond to certain boundary heights for the Gaussian free field.