In this paper, we present electrooptic experiments on photonic crystal fibers filled with a liquid crystalline blue phase. These fibers guide light via photonic band gaps (PBGs). The blue phase is isotropic in the field-off state but becomes birefringent under an electric field. This leads to a polarization dependent shift of the PBGs. Interestingly, the effect on the PBGs is asymmetrical: while the short wavelength edges of the PBGs shift, the long wavelength edges are almost unaffected. By performing band gap and modal analyses via the finite element simulations, we find that the asymmetric shift is the result of the mixed polarization of the involved photonic bands. Finally, we use the band gap shifts to calculate effective Kerr constants of the blue phase.