Dark conglomerates of domains with opposite handedness, which are designated as dark conglomerate phases (DC phases), have attracted much attention. After designing an achiral liquid crystal trimer, 4,4′-bis{7-4-(5-octyloxypyrimidin-2-yl)phenyloxyheptyloxy}biphenyl (1), which exhibits only a nematic phase, we prepared binary mixtures with some typical rod-like nematic liquid crystals, i.e., 4′-hexyloxy-4-cyanobiphenyl (6OCB), 2-(4-hexyloxyphenyl)-5-pentyloxypyrimidine (PPY), or 4-methyloxyphenyl 4-hexyloxycyclohexanecarboxylate (PCA), and investigated their phase transition behaviour. The binary mixtures containing 55–90 mol% of 6OCB were found to exhibit a nematic phase and a DC phase of chiral domains with opposite handedness. However, neither PPY nor PCA induced such a chiral conglomerate phase in the mixture with trimer 1. We discuss how core–core interactions contribute to produce such a chiral conglomerate phase.