This paper introduces a round-robin approach for multi-source localization based on matched-field processing. Each new source location is estimated from the ambiguity function after nulling from the data vector the current source location estimates using a robust projection matrix. This projection matrix effectively minimizes mean-square energy near current source location estimates subject to a rank constraint that prevents excessive interference with sources outside of these neighborhoods. Numerical simulations are presented for multiple sources transmitting through a fixed (and presumed known) generic Pekeris ocean waveguide in the single-frequency and broadband-coherent cases that illustrate the performance of the proposed approach which compares favorably against other previously published approaches. Furthermore, the efficacy with which randomized back-propagations may also be incorporated for computational advantage is also presented.