The TROPOspheric Monitoring Instrument (TROPOMI), aboard the Sentinel-5 Precursor (S5P) satellite launched in October 2017, is dedicated to monitoring the atmospheric composition associated with air quality and climate change. This paper presents the global retrieval of TROPOMI tropospheric formaldehyde (HCHO) and nitrogen dioxide (NO2) vertical columns using an updated version of the Peking University OMI NO2 (POMINO) algorithm, which focuses on improving the calculation of air mass factors (AMFs). The algorithm features explicit corrections for the surface reflectance anisotropy and aerosol optical effects, and it uses daily high-resolution (0.25°×0.25°) a priori HCHO and NO2 profiles from the Global Earth Observing System Composition Forecast (GEOS-CF) dataset. For cloud correction, a consistent approach is used for both HCHO and NO2 retrievals, where (1) the cloud fraction is recalculated at 440 nm using the same ancillary parameters as those used in the NO2 AMF calculation, and (2) the cloud-top pressure is taken from the operational FRESCO-S cloud product. The comparison between POMINO and reprocessed (RPRO) operational products in April, July and October 2021 as well as January 2022 exhibits high spatial agreement, but RPRO tropospheric HCHO and NO2 columns are lower by 10 % to 20 % over polluted regions. Sensitivity tests with POMINO show that the HCHO retrieval differences are mainly caused by different aerosol correction methods (implicit versus explicit), prior information on vertical profile shapes and background corrections, while the NO2 retrieval discrepancies result from different aerosol corrections, surface reflectances and a priori vertical profile shapes as well as their nonlinear interactions. With explicit aerosol corrections, the HCHO structural uncertainty due to the cloud correction using different cloud parameters is within ±20 %, mainly caused by cloud height differences. Validation against ground-based measurements from global Multi-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) observations and the Pandonia Global Network (PGN) shows that in April, July and October 2021 as well as in January 2022 POMINO retrievals present a comparable day-to-day correlation but a reduced bias (normalized mean bias, NMB) compared to the RPRO products (HCHO: R=0.62, NMB = -30.8% versus R=0.68, NMB = -35.0%; NO2: R=0.84, -9.5% versus R=0.85, -19.4%). An improved agreement of the HCHO / NO2 ratio (FNR, formaldehyde to nitrogen dioxide ratio) with MAX-DOAS and PGN measurements based on POMINO retrievals is also found (NMB: −14.8 % versus −21.1 %). Our POMINO retrieval provides a useful source of information, particularly for studies combining HCHO and NO2.