We introduce a new class of non-uniformly correlated beams that are called rectangular Hermite non-uniformly correlated (RHNUC) beams, which possess rectangular symmetry in their degree of coherence. It is shown that, in free space and in turbulence, these beams possess self-focusing properties and that the position of the focus can be adjusted in 3-D space by manipulating the correlation properties of the source. Furthermore, it is demonstrated that, by choosing different mode orders and correlation lengths along two transverse directions, one creates astigmatic beams that can be designed to have a high and near-constant intensity over an extended propagation range. (C) 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement