  We present APEX-LABOCA 870 micron observations of the fields surrounding the
nine brightest, high-redshift, unlensed objects discovered in the South Pole
Telescope's (SPT) 2500 square degrees survey. Initially seen as point sources
by SPT's 1-arcmin beam, the 19-arcsec resolution of our new data enables us to
deblend these objects and search for submillimetre (submm) sources in the
surrounding fields. We find a total of 98 sources above a threshold of 3.7
sigma in the observed area of 1300 square arcminutes, where the bright central
cores resolve into multiple components. After applying a radial cut to our
LABOCA sources to achieve uniform sensitivity and angular size across each of
the nine fields, we compute the cumulative and differential number counts and
compare them to estimates of the background, finding a significant overdensity
of approximately 10 at 14 mJy. The large overdensities of bright submm sources
surrounding these fields suggest that they could be candidate protoclusters
undergoing massive star-formation events. Photometric and spectroscopic
redshifts of the unlensed central objects range from 3 to 7, implying a volume
density of star-forming protoclusters of approximately 0.1 per giga-parsec
cube. If the surrounding submm sources in these fields are at the same
redshifts as the central objects, then the total star-formation rates of these
candidate protoclusters reach 10,000 solar masses per year, making them much
more active at these redshifts than what has been seen so far in both
simulations and observations.
