We present an extensive ALMA spectroscopic follow-up programme of the z = 4.3 structure SPT2349-56, one of the most actively star-forming protocluster cores known, to identify additional members using their [C] 158 mu m and CO(4-3) lines. In addition to robustly detecting the 14 previously published galaxies in this structure, we identify a further 15 associated galaxies at z = 4.3, resolving 55 +/- 5 per cent of the 870 mu m flux density at 0.5 arcsec resolution compared to 21 arcsec single-dish data. These galaxies are distributed into a central core containing 23 galaxies extending out to 300 kpc in diameter, and a northern extension, offset from the core by 400 kpc, containing three galaxies. We discovered three additional galaxies in a red Herschel-SPIRE source 1.5 Mpc from the main structure, suggesting the existence of many other sources at the same redshift as SPT2349-56 that are not yet detected in the limited coverage of our data. An analysis of the velocity distribution of the central galaxies indicates that this region may be virialized with a mass of (9 +/- 5) x 10(12) M-circle dot, while the two offset galaxy groups are about 30 and 60 per cent less massive and show significant velocity offsets from the central group. We calculate the [CI] and far-infrared number counts, and find evidence for a break in the [CII] luminosity function. We estimate the average SFR density within the region of SPT2349-56 containing single-dish emission (a proper diameter of 720 kpc), assuming spherical symmetry, to be roughly 4 x 10(4) M-circle dot yr(-1) Mpc(-3); this may be an order of magnitude greater than the most extreme examples seen in simulations.