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The XOR-Ising model on a graph consists of random spin configurations onvertices of the graph obtained by taking the product at each vertex of thespins of two independent Ising models. In this paper, we explicitly relate loopconfigurations of the XOR-Ising model and those of a dimer model living on adecorated, bipartite version of the Ising graph. This result is proved forgraphs embedded in compact surfaces of genus g.  Using this fact, we then prove that XOR-Ising loops have the same law aslevel lines of the height function of this bipartite dimer model. Atcriticality, the height function is known to converge weakly in distribution toa Gaussian free field.  As a consequence, results of this paper shed a light on the occurrence of theGaussian free field in the XOR-Ising model. In particular, they prove adiscrete analogue of Wilson's conjecture, stating that the scaling limit ofXOR-Ising loops are "contour lines" of the Gaussian free field.

Height representation of XOR-Ising loops via bipartite dimers