Fully-connected beamforming networks (FCBNs) can provide better performance in a hybrid beamforming (HBF) architecture for massive multiple-input-multiple-output (MIMO) systems. The traditional FCBN adopts the "splitters + phase shifters + attenuators + combiners" architecture, which has a huge intrinsic loss by the theoretical analysis. The network loss and topology complexity pose challenges for its engineering applications. To address the aforementioned issues, this article proposes a new efficient FCBN architecture, eliminating the intrinsic loss of the network. The proposed new FCBN architecture consists of tunable and fixed directional couplers and phase shifters in cascading, achieving both amplitude and phase adjustments. For synthesizing such a tunable network with arbitrary I/O ports, the complete QR factorization synthesis method with the corresponding tunable complex Givens rotation matrix has been proposed and proved. To demonstrate the proposed synthesis method, a 2x4 FCBN network has been synthesized, fabricated, and measured. Concentrating on the characteristics of the FCBN, an optimization process has been introduced to simplify hardware topology complexity. The results show that the proposed FCBN architecture has advantages in no intrinsic loss, simple topology, and low cost, making it widely applied in wireless communication systems to improve the system efficiency.