The primary sources of auxiliary data required for TROPOMI data processing include ECMWF meteorological fields, Suomi NPP VIIRS cloud information, NSIDC NISE snow and ice cover data, and KNMI TM5 model output for the NO2 data product. A detailed description of all auxiliary data can be found here in the TROPOMI Input/Output Data Definition (IODD) document.
Links for the auxiliary datasets:
ECMWF - European Centre for Medium Weather Forecasting
VIIRS - Visible Infrared Imaging Radiometer Suite on board Suomi NPP (National Polar-Orbiting Partnership)
NISE - Near real-time Ice and Snow Extent from the National Snow & Ice Data Center (NSIDC)
TM5 - Global Chemistry Transport Model
This figure from the S5P TROPOMI IODD document depicts horizontal gridding (reduced Gaussian N640) of ECMWF data used by TROPOMI data algorithms, where the green square in the upper left corner (10x10km) shows the relative scale of a TROPOMI pixel.