We report a high-field thermodynamic study of the hyperhoneycomb Kitaev material β-Li₂IrO₃, using magnetotropic susceptibility to resolve its low-temperature field-angle phase diagram across the principal crystallographic planes in magnetic fields up to 60 T. Rather than evolving directly from the low-field incommensurate state into a polarized regime, the system exhibits a strongly direction-dependent sequence of correlated phases. Most notably, for fields in the ac-plane, we identify an additional high-field phase that is absent in the other principal planes and exists only within a restricted region of field-angle space. This phase structure is naturally explained by the competition between magnetic field and bond-directional exchange interactions. Using a symmetry-based description supported by microscopic calculations within the J-K-Γ model, we show that off-diagonal Γ exchange couples the ferromagnetic and staggered magnetic orders and thereby stabilizes the observed correlated high-field phases. The measured angular dependence of the critical fields is quantitatively captured by this theory, identifying Γ exchange as the key interaction controlling the high-field response. These results clarify why the promise of a field-induced spin liquid – the notion that suppressing magnetic order might reveal the underlying Kitaev physics – remains unfulfilled in candidate materials: even when the Kitaev interaction is large, off-diagonal exchange stabilizes symmetry-constrained correlated phases that instead preempt the polarized state.