We identify, for the first time to our best knowledge, a new type of transmission band having hybrid resonance nature in hollow-core anti-resonant fibers (ARF). We elucidate its unique phase-locking feature of the electric field at the outermost boundary. Exploiting this hybrid band, large birefringence in the order of 10(-4) is obtained. Our analyses based on Kramer-Kronig relation and transverse field confinement interpret the link between the hybrid transmission band and the large birefringence. Guided by these analyses, an experimentally realizable polarization-maintaining ARF design is proposed by introducing multi-layered dielectric structure into a negative curvature core-surround. This multi-layered ARF possesses characteristics of low loss, broad transmission band and large birefringence simultaneously.