The recent measurement of the antihydrogen terrestrial gravitational acceleration with the ALPHA-g apparatus at CERN relies upon the detection of the annihilation of the anti-atoms that are released from their magnetic confinement and that move under the influence of gravity. The ALPHA-g magnetic trap is surrounded by a Time Projection Chamber designed to identify the annihilation products and to reconstruct the annihilation position. The TPC is called “radial”, or rTPC, because the drift field is perpendicular to the trap axis and, therefore, to the magnetic field of the external field. The design, construction and commissioning of this detector are described in the following.