The idea is to send light through a beam splitter and one part goes through a right circular polariser and the other goes through a left circular polariser. Will the emergent beams still be in phase?
We have to types of coherency. Temporal and spatial until you don't change beam size and apply spatial filters, the spatial coherency doesn't change.
In the case of temporal coherency, the circular polarizer only apply a 90 degree phase delay between $E_x$ and $E_y$. It doesn't change the Statistic properties of light. So the Temporal coherency remains unchanged.
Let us follow the lights path step by by.
The beam of a perfect laser is coherent because all photons are generated with the same wavelength (stimulated emission) and leave the laser cavity through a half mirrored end, all in the same phase. But in such a “pure” laser the photons are not polarized.
Let the beam splitter be made by two half prisms with a small air gap between them Glan-Taylor prism. The incoming beam is devised now into two polarized beams. As commented by John Custer, the path lengths are important for the spatial coherence. The numbers of wave periodes in the prism as well as the numbers of wave periodes from there to the unification point determines the spatial coherence.
Adding circular polarizers with identical path lengths (same thickness and same material properties) does not change the spatial coherence but will destroy the linear polarizations of the beams.
Will the emergent beams still be in phase?
Under the described above circumstances, yes, the beams with their crests still are at the identical positions after the beams merged together again.