We introduce a novel iterative method for solving nonlinear inverse scattering problems. Inspired by the theory of nonlocality, we formulate the inverse scattering problem in terms of reconstructing the nonlocal unknown scattering potential V from scattered field measurements made outside a sample. Utilizing the one-to-one correspondence between V and T, the T-matrix, we iteratively search for a diagonally dominated scattering potential V corresponding to a data compatible T-matrix T. This formulation only explicitly uses the data measurements when initializing the iterations, and the size of the data set is not a limiting factor. After introducing this method, named data-compatible T-matrix completion (DCTMC), we detail numerous improvements the speed up convergence. Numerical simulations are conducted that provide evidence that DCTMC is a viable method for solving strongly nonlinear ill-posed inverse problems with large data sets. These simulations model both scalar wave diffraction and diffuse optical tomography in three dimensions. Finally, numerical comparisons with the commonly used nonlinear iterative methods Gauss-Newton and Levenburg-Marquardt are provided.