We consider a model for a one-dimensional quantum wire with Rashba spin-orbit coupling and repulsive interactions, proximity coupled to a conventional s-wave superconductor. Using a combination of Hartree-Fock and density matrix renormalization group calculations, we show that for sufficiently strong interactions in the wire, a time-reversal invariant topological superconducting phase can be stabilized in the absence of an external magnetic field. This phase supports two zero-energy Majorana bound states at each end, which are protected by time-reversal symmetry. The mechanism for the formation of this phase is a reversal of the sign of the effective pair potential in the wire, due to the repulsive interactions. We calculate the differential conductance into the wire and its dependence on an applied magnetic field using the scattering-matrix formalism. The behavior of the zero-bias anomaly as a function of the field direction can serve as a distinct experimental signature of the topological phase.