We study a liquid of zigzagging two-dimensional directed polymers with bending rigidity, i.e., polymers whose conformations follow checkerboard paths. In the continuum limit the statistics of such polymers obey the Dirac equation for particles of imaginary mass. We exploit this observation to investigate a liquid of these polymers via a quantum many-fermion analogy. A self-consistent approximation predicts a phase of tilted order, in which the polymers may develop a preference to zig rather than zag. We compute the phase diagram and key response functions for the polymer liquid, and comment on the role played by fluctuations.