Given two Fuchsian representations ρ l and ρr of the fundamental group of a closed oriented surface S of genus ≥ 2, we study the relation between Lagrangian submanifolds of Mρ = (H^2/ρ_l(π_1(S))) × (H^2/ρ_r(π_1(S))) and ρ-equivariant embeddings σ of S into Anti-de Sitter space, where ρ = (ρ_l ,ρ_r) is the corresponding representation into PSL(2,R) × PSL(2,R). It is known that, if σ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold Λ ML of Mρ. We show that, given any ρ-equivariant embedding σ, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to Λ_ML. Conversely, any Lagrangian submanifold Hamiltonian isotopic to Λ_ML is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.