Conventional hybrid beamforming architectures are often compared with one another and with the fully-digital architecture under the same \emph{radiated} antenna power. However, the physically relevant budget is the power injected by the RF-chain outputs into the passive analog RF network, which is then usually transferred to the antenna ports in a contractive (lossy) manner. This issue is especially pronounced for fully-connected splitter--phase-shifter--combiner networks, whose physical power transfer remains contractive even under ideal passive-component assumptions. Motivated by this injected-power viewpoint, this paper proposes a hybrid beamforming architecture based on a programmable unitary RF network. Under ideal passive-component assumptions, all injected RF-chain power reaches the antenna ports without loss. The analog RF network is realized as an \emph{interlaced mixer--phase} architecture consisting of fixed (non-programmable) mixing layers interleaved with programmable diagonal phase-shifting layers. We derive a closed-form digital beamformer and a low-complexity programming method for the analog beamformer, yielding a hybrid precoder that closely matches the fully-digital precoder. Narrowband simulations with continuous and quantized phases, benchmarked against the fully-digital architecture, the physically modeled fully-connected phase-shifter baselines, and an ideal-lossless Butler/DFT beam-selection baseline under equal total injected RF-chain power, show that the continuous-phase and 6-bit realizations of the proposed architecture are nearly indistinguishable from the fully-digital benchmark and achieve significant gains over the baseline hybrid beamforming architectures.