We study the ground-state and low-lying metastable phases of repulsive binary Bose-Einstein condensates confined in twisted, spin-dependent periodic optical lattices. For balanced mixtures, weak intercomponent interactions yield a fourfold momentum-space symmetry dictated by the lattice geometry. Increasing the coupling strength leads to the emergence of additional momentum peaks that combine with the lattice-induced structure to produce an eightfold rotationally symmetric pattern, signaling quasiperiodic order. At intermediate interactions, global phase separation suppresses this quasicrystalline state; however, at stronger coupling, local phase separation gives rise to a long-lived metastable phase in which the eightfold symmetry is restored. In this regime, a secondary ring of dominant momentum peaks appears at smaller wave vectors, indicating longer-wavelength density modulations and a crossover from lattice-dominated to interaction-driven quasicrystalline order. In contrast, imbalanced mixtures form partially miscible density clusters with eightfold-symmetric quasiperiodic patterns only at intermediate coupling, while stronger interactions drive global phase separation and permanently destroy quasicrystalline order. Real-time simulations demonstrate that these quasiperiodic structures are dynamically stable and experimentally accessible. Our results show that quasicrystalline order can emerge in binary condensates without explicitly quasiperiodic lattices and reveal population balance as a key ingredient for stabilizing quantum quasicrystals.

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• Bose-Bose mixtures
• Bose-Einstein condensates
• Cold atoms & matter waves
• Cold gases in optical lattices
• Collective effects in atomic physics