Context. The majority of massive stars are born with a close binary companion. How this affects their evolution and fate is still largely uncertain, especially at low metallicity. Aims. We derive synthetic populations of massive post-interaction binary products and compare them with corresponding observed populations in the Small Magellanic Cloud (SMC). Methods. We analyse 53298 detailed binary evolutionary models computed with MESA. Our models include the physics of rotation, mass and angular momentum transfer, magnetic internal angular momentum transport, and tidal spin-orbit coupling. They cover initial primary masses of 5─100 M⊙, initial mass ratios of 0.3─0.95, and all initial periods for which interaction is expected, 1─3162 d. They are evolved through the first mass transfer and the donor star death, and a a possible ensuing Be X-ray binary phase, and they end when the mass gainer leaves the main sequence. Results. In our fiducial synthetic population, 8% of the OB stars in the SMC are post-mass-transfer systems, and 7% are merger products. In many of our models, the mass gainers are spun up and expected to form Oe/Be stars. While our model underpredicts the number of Be X-ray binaries in the SMC, it reproduces the main features of their orbital period distribution and the observed number of SMC binary WR stars. We further expect ∼50 OB+BH binaries below and ∼170 above the 20 d orbital period. The long-period OB+BH binaries might produce merging double black holes. However, their progenitors, the predicted long-period WR+OB binaries, are not observed. Conlcusions. While the comparison with the observed SMC stars supports many physics assumptions in our high-mass binary models, a better match for the large number of observed OBe stars and Be X-ray binaries likely requires a lower merger rate and/or a higher mass transfer efficiency during the first mass transfer. The fate of the initially wide O star binaries remains particularly uncertain.