We report the detection of ADFS-27, a dusty, starbursting major merger at a redshift of z = 5.655, using the Atacama Large Millimeter/submillimeter Array (ALMA). ADFS-27 was selected from Herschel/Spectral and Photometric Imaging Receiver (SPIRE) and APEX/LABOCA data as an extremely red “870 μm riser” (i.e., {S}250μ {{m}}< {S}350μ {{m}}< {S}500μ {{m}}< {S}870μ {{m}}), demonstrating the utility of this technique to identify some of the highest-redshift dusty galaxies. A scan of the 3 mm atmospheric window with ALMA yields detections of CO(J = 5 → 4) and CO(J = 6 → 5) emission, and a tentative detection of H2O(211 → 202) emission, which provides an unambiguous redshift measurement. The strength of the CO lines implies a large molecular gas reservoir with a mass of M gas = 2.5 × 1011 ({α }{CO}/0.8)(0.39/{r}51) M ⊙, sufficient to maintain its ∼2400 M ⊙ yr‑1 starburst for at least ∼100 Myr. The 870 μm dust continuum emission is resolved into two components, 1.8 and 2.1 kpc in diameter, separated by 9.0 kpc, with comparable dust luminosities, suggesting an ongoing major merger. The infrared luminosity of L IR ≃ 2.4 × 1013 L ⊙ implies that this system represents a binary hyper-luminous infrared galaxy, the most distant of its kind presently known. This also implies star formation rate surface densities of {{{Σ }}}{SFR}=730 and 750 M ⊙ yr‑1 kpc2, consistent with a binary “maximum starburst.” The discovery of this rare system is consistent with a significantly higher space density than previously thought for the most luminous dusty starbursts within the first billion years of cosmic time, easing tensions regarding the space densities of z ∼ 6 quasars and massive quiescent galaxies at z ≳ 3.