In laser beam projection through the atmosphere under conditions where strong scintillation is present, real zeros can appear in the beacon field. Recent advances in phase reconstruction have enabled phase estimation from phase differences in the presence of branch points, or real zeros, in the beacon field. While it is not possible to perfectly fit a physically realizable deformable mirror to the resulting discontinuous phase, the errors in the deformable mirror figure can be minimized based on knowledge of the locations and general shape of the discontinuities. In this paper we review two approaches to reconstructing phase in the presence of branch points, and compare the performance of these approaches to each other and to a conventional least-squares phase reconstructor in a laser beam projection system.© (1999) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.