Localizing near-field sources considering practical arrays is a recent challenging topic for next generation wireless communication systems. Practical antenna array apertures with closely spaced elements exhibit direction-dependent mutual coupling (MC), which can significantly degrade the performance localization techniques. A conventional method for near-field localization in the presence of MC is the three-dimensional (3D) multiple signal classification technique, which, however, suffers from extremely high computational complexity. Recently, two-dimensional (2D) search alternatives have been presented, exhibiting increased complexity still for direction-dependent MC scenarios. In this paper, we devise a low complexity one-dimensional (1D) iterative method based on an oblique projection operator (IMOP) that estimates direction-dependent MC and the locations of multiple near-field sources. The proposed method first estimates the initial direction of arrival (DOA) and MC using the approximate wavefront model, and then, estimates the initial range of one near-field source using the exact wavefront model. Afterwards, at each iteration, the oblique projection operator is used to isolate components associated with one source from those of other sources. The DOA and range of this one source are estimated using the exact wavefront model and 1D searches. Finally, the direction-dependent MC is estimated for each pair of the estimated DOA and range. The performance of the proposed near-field localization approach is comprehensively investigated and verified using both a full-wave electromagnetic solver and synthetic simulations. It is showcased that our IMOP scheme performs almost similarly to a state-of-the-art approach but with a 42 times less computational complexity.