We present data processing and verification of the Southern Twenty-centimetre All-sky Polarization Survey (STAPS) conducted with Murriyang, the Parkes 64-m telescope. The survey covers the sky area of -89{deg}<Dec<0{deg} and the frequency range of 1.3-1.8GHz split into 1-MHz channels. STAPS was observed commensally with the S-band Polarization All-Sky Survey (S-PASS). The survey is composed of long azimuth scans, which allows us to absolutely calibrate Stokes Q and U with the data processing procedure developed for S-PASS. We obtained I, Q, and U maps on both the flux density scale (Jy/beam) and the main beam brightness temperature scale (K), for the 301 frequency channels with sufficiently good data. The temperature scale is tied to the Global Magneto-ionic Medium Survey (GMIMS) high-band north sky survey conducted with the Dominion Radio Astrophysical Observatory 26-m telescope. All the STAPS maps are smoothed to a common resolution of 20'. The root mean square (rms) noise per channel ranges from about 16mK to 8mK for I, and from about 8mK to 5mK for Q and U, at frequencies from 1.3 to 1.8GHz. The rms noise in Q and U varies with declination and reaches minimum at declination of -89{deg}. We also ran rotation measure (RM) synthesis and RM clean to obtain peak polarized intensity and Faraday depth maps. The whole STAPS data processing was validated by comparing flux densities of compact sources, pixel flux density versus pixel flux density for Cen A, pixel temperature versus pixel temperature for the entire survey area, and the RMs of extragalactic sources between STAPS and other measurements. The uncertainty of the flux density scale is less than 10%. STAPS delivers an L-band ({lambda}20cm) multifrequency polarization view of the Galaxy, and will help advance our understanding of the Galactic magnetic field and magnetized interstellar medium.