GaAsPBi is a new class of quaternary III-V compounds that extends the concept of band gap engineering on GaAs with potentials for lattice matching and excellent temperature stability. The alloy has so far been grown only by metalorganic vapor phase epitaxy and this work represents the first epitaxial results of the alloy grown by molecular beam epitaxy (MBE), an alternative technique and better suited for low-temperature processes involving Bismuth. Crystalline quality of the alloys is probed by high-resolution x-ray diffraction and photoluminescence (PL) which show that smooth and optically active films can be grown in limited parameter windows. Temperature-dependent PL shows that the 200 nm, MBE-grown GaAs 0.38 P 0.44 Bi 0.18 film (the composition estimated using x-ray photoelectron spectroscopy) has a band gap temperature stability close to that of GaAsBi, and superior to GaAs. The role of Bi in the quaternary alloy is complicated: Bi not only gets incorporated into the growing film but also enhances the P molar fraction. Based on this insight, strategies for growing GaAsPBi epilayers which are lattice-matched to GaAs are described, potentially affecting many important III-V based heterostructures such as lasers, light-emitting diodes, and solar cells.