Galaxy submillimetre number counts are a fundamental measurement in our understanding of galaxy evolution models. Most early measurements are obtained via single-dish telescopes with substantial source confusion, whereas recent interferometric observations are limited to small areas. We used a large database of ALMA continuum observations to accurately measure galaxy number counts in multiple (sub)millimetre bands, thus bridging the flux density range between single-dish surveys and deep interferometric studies. We continued the Automated Mining of the ALMA Archive in the COSMOS Field project (A3COSMOS) and extended it with observations from the GOODS-South field (A3GOODSS). The database consists of ~4,000 pipeline-processed continuum images from the public ALMA archive, yielding 2,050 unique detected sources. To infer galaxy number counts, we constructed a method to reduce the observational bias inherent to targeted pointings that dominate the database. This method comprises a combination of image selection, masking, and source weighting. The effective area was calculated by accounting for inhomogeneous wavelengths, sensitivities, and resolutions and for spatial overlap between images. We tested and calibrated our method with simulations. We obtained the first number counts derived in a consistent and homogeneous way in four different ALMA bands covering a relatively large area. The results are consistent with number counts from the literature within the uncertainties. We extended the available depth in ALMA Band 4 by 0.4 dex with respect to previous studies. In Band 7, at the depth of the inferred number counts, ~40% of the cosmic infrared background is resolved into discrete sources. This fraction, however, decreases with wavelength, reaching ~4% in Band 3. Finally, we used the number counts to test models of dusty galaxy evolution, and find a good agreement within the uncertainties.