  We theoretically study the nematic ordering transition of rods that are able
to elastically adjust their mutually excluded volumes. The model rods, which
consist of a hard core surrounded by a deformable shell, mimic the structure of
polymer-coated, rod-like fd virus particles that have recently been the object
of experimental study [K.~Purdy et al., Phys. Rev. Lett. \textbf{94}, 057801
(2005)]. We find that fluids of such soft rods exhibit an isotropic-nematic
phase transition at a density higher than that of the corresponding hard-rod
system of identical diameter, and that at coexistence the order parameter of
the nematic phase depends non monotonically on the elastic properties of the
polymer coating. For binary mixtures of hard and soft rods, the topology of the
phase diagram turns out to depend sensitively on the elasticity of shell. The
lower nematic-nematic critical point, discovered in mixtures of bare and
polymer-coated fd virus particles, is not reproduced by the theory.
