We define a new Edwards-Sokal representation of the Ising model using double random currents, and introduce a joint coupling along with a XOR-Ising model and the associated height function. After taking the scaling limit of all discrete structures, we discuss our main result: a natural coupling of the Ising magnetisation field and the Gaussian free field. To the best of our knowledge, the existence of such a coupling was not predicted previously, and extends the bosonization picture for the XOR-Ising model. Moreover, the coupling has a geometric nature and is built via the two-valued sets of the GFF. We will touch on the main elements of the proof: an FKG property of the relevant percolation model, the resulting RSW theory, and an L^2 approximation scheme for the counting (discrete area) measures on the clusters of the percolation. As a challenge for the future, we also state related conjectures for the Ashkin-Teller model.