We study the structure of $l=1$ modes of strongly anisotropic coiled weakly guiding optical fibers. By solving the vector wave equation within the framework of the perturbation theory with degeneracy, we analytically establish the expressions for modes and their polarization corrections. We show that, at certain parameters of the fiber helix, the $l=1$ modes are represented by almost pure optical vortices that maintain a linear polarization in the Frenet frame. We demonstrate that, in this case, the propagation constants comprise geometrically induced terms that are proportional to the orbital angular momentum (OAM) of the mode. We show that the vortex modes acquire upon propagation additional topological phases proportional to their intrinsic OAM and to the solid angle subtended by one helix coil. The presence of such a topological phase results in rotation (at a constant polarization) of the intensity patterns; after one coil the rotation angle equals the solid angle subtended by a coil.