This paper presents the Forward Time Projection Chamber (FTPC) system of the NA61/SHINE experiment at the CERN SPS accelerator. This TPC system applies a novel tandem-TPC design to reduce the background originating from particle tracks not synchronous with the event trigger. The FTPC system is composed of three chambers with alternating drift field directions. The chambers were installed directly along the beamline region of the NA61/SHINE detector in a medium- to high-intensity (10−100 kHz) hadron or ion beam. The tandem TPC system has proved to be capable of rejecting out-of-time background tracks not associated with a primary interaction. In addition, the system performs tracking and inclusive dE/dx particle identification for particles at and near the beam momentum. This shows that a tandem-TPC-based chamber design may be used also in other experimental applications with a demand for low material budget, tracking capability, and the need for dE/dx particle identification, all while in the presence of a relatively high particle flux.