Strong, long-range dipole–dipole interactions between interlayer excitons (IXs) can lead to new multiparticle correlation regimes1,2, which drive the system into distinct quantum and classical phases2–5, including dipolar liquids, crystals and superfluids. Both repulsive and attractive dipole–dipole interactions have been theoretically predicted between IXs in a semiconductor bilayer2,6–8, but only repulsive interactions have been reported experimentally so far3,9–16. This study investigated free-standing, twisted (51°, 53°, 45°) tungsten diselenide/tungsten disulfide (WSe2/WS2) heterobilayers, in which we observed a transition in the nature of dipolar interactions among IXs, from repulsive to attractive. This was caused by quantum-exchange-correlation effects, leading to the appearance of a robust interlayer biexciton phase (formed by two IXs), which has been theoretically predicted6–8 but never observed before in experiments. The reduced dielectric screening in a free-standing heterobilayer not only resulted in a much higher formation efficiency of IXs, but also led to strongly enhanced dipole–dipole interactions, which enabled us to observe the many-body correlations of pristine IXs at the two-dimensional quantum limit. In addition, we firstly observed several emission peaks from moiré-trapped IXs at room temperature in a well-aligned, free-standing WSe2/WS2 heterobilayer. Our findings open avenues for exploring new quantum phases with potential for applications in non-linear optics.