Geometric frustration gives rise to new fundamental phenomena and is known to yield the formation of exotic states of matter, such as incommensurate crystals, modulated liquid-crystalline phases, and phases stabilized by defects. In this Letter, we present a detailed study of polar structure of freely suspended fluid filaments in a polarization modulated liquid-crystal phase. We show that a periodic pattern of polarization-splay stripes separated by defect boundaries and decorating smectic layers can stabilize the structure of fluid fibers against the Rayleigh-Plateau instability. The instability is suppressed by the resistance of the defect structure to a radial compression of the cylindrical fibers. Our results provide direct experimental observation of a link between the stability of the liquid fibers, internal polar order, and geometrical constraints. They open a new perspective on a wide range of fluid polar fiber materials.