We demonstrate that exceptional points exist in fully transparent, optically ”effectively” biaxial, anisotropic micro-cavities, fabricated using an uniaxial cavity material with its axis inclined to the Bragg mirror growth direction. This is similar to the existence of singular (optic) axes in absorbing biaxial crystals, but the lack of time reversal symmetry is mediated by the mode broadening, i.e. the photon escape from the – in principle – open cavity system. As a consequence the eigenmodes are generally elliptically polarized, and completely circularly polarized eigenmodes are expected in certain directions. Via geometric and chemical composition design degrees of freedom, the spectral and angular position of these chiral modes can be rationally designed. Possible applications arise from the use of such directions for circularly polarized emission without the use of spin injection or internal or external magnetic fields. Also the coupling of such modes to excitons, adding oscillator strength to the system, seems a promising avenue of research.