The Ashkin-Teller model is a pair of interacting Ising models and has two parameters: J is a coupling constant in the Ising models and U describes the strength of the interaction between them. In the ferromagnetic case J,U>0 on the square lattice, we establish a complete phase diagram conjectured in physics in 1970s (by Kadanoff and Wegner, Wu and Lin, Baxter and others): when J We use a finite-criterion argument and continuity to extend the result of Peled and the third author \cite{GlaPel19} from a self-dual point to its neighborhood. Our proofs go through the random-cluster representation of the Ashkin-Teller model introduced by Chayes-Machta and Pfister-Velenik and we rely on couplings to FK-percolation.
