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Cloud shadows can be detected in the radiance measurements of the TROPOMI instrument on board the Sentinel-5P satellite due to its high spatial resolution, and could possibly affect its air quality products. The cloud shadow induced signatures are, however, not always apparent and may depend on various cloud and scene parameters. Hence, the quantification of the cloud shadow impact requires the analysis of large data sets. Here we use the cloud shadow detection algorithm DARCLOS to detect cloud shadow pixels in the TROPOMI absorbing aerosol index (AAI) product over Europe during 8 months. For every shadow pixel, we automatically select cloud- and shadow-free neighbour pixels, in order to estimate the cloud shadow induced signature. In addition, we simulate the measured cloud shadow impact on the AAI with our newly developed 3D radiative transfer algorithm MONKI. Both the measurements and simulations show that the average cloud shadow impact on the AAI is close to zero (0.06 and 0.16, respectively). However, the top-of-atmosphere reflectance ratio between 340 and 380 nm, which is used to compute the AAI, is significantly increased in 95 % of the shadow pixels. So, cloud shadows are bluer than surrounding non-shadow pixels. Our simulations explain that the traditional AAI formula intrinsically already corrects for this cloud shadow effect, via the lower retrieved scene albedo. This cancellation of cloud shadow signatures is not always perfect, sometimes yielding second order low and high biases in the AAI which we also successfully reproduce with our simulations. We show that the magnitude of those second order cloud shadow effects depends on various cloud parameters which are difficult to determine for the shadows measured with TROPOMI. We conclude that a potential cloud shadow correction strategy for the TROPOMI AAI would therefore be complicated if not unnecessary. [ABSTRACT FROM AUTHOR]