A new two field iterated Asymptotic-Preserving method is introduced for the numerical resolution of strongly anisotropic elliptic equations. This method does not rely on any integration of the field defining the anisotropy. It rather harnesses an auxiliary variable removing any stiffness from the equation. Compared to precedent realizations using the same approach, the iterated method allows for the resolution of each field independently within an iterative process to converge the two unknowns. This brings advantages in the computational efficiency of the method for large meshes, a better scaling of the matrices condition number with respect to the mesh refinement as well as the ability to address complex anisotropy topology including closed field lines.