We present an ALMA study of the similar to 180 brightest sources in the SCUBA-2 850-mu m map of the COSMOS field from the S2COSMOS survey, as a pilot study for AS2COSMOS - a full survey of the similar to 1000 sources in this field. In this pilot study, we have obtained 870-mu m continuum maps of an essentially complete sample of the brightest 182 sub-millimetre sources (S-850 mu m > 6.2 mJy) in COSMOS. Our ALMA maps detect 260 sub-millimetre galaxies (SMGs) spanning a range in flux density of S-870 mu m = 0.7-19.2 mJy. We detect more than one SMG counterpart in 34 +/- 2 percent of sub-millimetre sources, increasing to 53 +/- 8 percent for SCUBA-2 sources brighter than S-850 mu m > 12 mJy. We estimate that approximately one-third of these SMG-SMG pairs are physically associated (with a higher rate for the brighter secondary SMGs, S-870 mu m greater than or similar to 3 mJy), and illustrate this with the serendipitous detection of bright [C II] 157.74-mu m line emission in two SMGs, AS2COS 0001.1 and 0001.2 at z = 4.63, associated with the highest significance single-dish source. Using our source catalogue, we construct the interferometric 870-mu m number counts at S-870 mu m > 6.2 mJy. We use the extensive archival data of this field to construct the multiwavelength spectral energy distribution of each AS2COSMOS SMG, and subsequently model this emission with MAGPHYS to estimate their photometric redshifts. We find a median photometric redshift for the S-870 mu m > 6.2 mJy AS2COSMOS sample of z = 2.87 +/- 0.08, and clear evidence for an increase in the median redshift with 870-mu m flux density suggesting strong evolution in the bright end of the 870-mu m luminosity function.