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We present BIFROST, an extended version of the GPU-accelerated hierarchical fourth-order forward symplectic integrator code FROST. BIFROST (BInaries in FROST) can efficiently evolve collisional stellar systems with arbitrary binary fractions up to $f_\mathrm{bin}=100\%$ by using secular and regularised integration for binaries, triples, multiple systems or small clusters around black holes within the fourth-order forward integrator framework. Post-Newtonian (PN) terms up to order PN3.5 are included in the equations of motion of compact subsystems with optional three-body and spin-dependent terms. PN1.0 terms for interactions with black holes are computed everywhere in the simulation domain. The code has several merger criteria (gravitational-wave inspirals, tidal disruption events and stellar and compact object collisions) with the addition of relativistic recoil kicks for compact object mergers. We show that for systems with $N$ particles the scaling of the code remains good up to $N_\mathrm{GPU} \sim 40\times N / 10^6$ GPUs and that the increasing binary fractions up to 100 per cent hardly increase the code running time (less than a factor $\sim 1.5$). We also validate the numerical accuracy of BIFROST by presenting a number of star clusters simulations the most extreme ones including a core collapse and a merger of two intermediate mass black holes with a relativistic recoil kick.