We identify and investigate the nature of the 20 brightest 250-micron sources detected by the BLAST within the central 150 sq. arcmin of the GOODS-South field. Aided by the available deep VLA radio imaging, reaching S_1.4 = 30 micro-Jy, we have identified radio counterparts for 17/20 of the 250-micron sources. The resulting enhanced positional accuracy of ~1 arcsec has then allowed us to exploit the deep multi-frequency imaging of GOODS-South to establish secure galaxy counterparts for the 17 radio-identified sources, and plausible galaxy candidates for the 3 radio-unidentified sources. Confusion is a serious issue for this deep BLAST 250-micron survey, due to the large size of the beam. Nevertheless, we argue that our chosen counterparts are significant, and often dominant contributors to the measured BLAST flux densities. For all of these 20 galaxies we have been able to determine spectroscopic (8) or photometric (12) redshifts. The result is the first `complete redshift distribution for a deep 250-micron selected galaxy sample. This reveals that 250-micron surveys reaching detection limits of ~30 mJy contain not only low-redshift spirals/LIRGs, but also the extreme z~2 dust-enshrouded starburst galaxies previously discovered at sub-millimetre wavelengths. Based on their IRAC colours, we find that virtually all of the BLAST galaxy identifications appear better described as analogues of the M82 starburst galaxy, or Sc star-forming discs rather than highly obscured ULIRGs. Inspection of the LABOCA 870-micron imaging of the GOODS-South field yields detections of 7/11 of the z>1 BLAST sources, and reveals 250/870 flux-density ratios consistent with a standard 40K modified black-body fit with a dust emissivity index beta=1.5.
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