For a coded massive multi-user multiple-input multiple-output (MIMO) system, a soft-output MIMO detector is essential since it can provide a significant coding gain, e.g., 3 dB, compared with a hard-output detector. However, the computational complexity of the soft-output MIMO detector is usually much greater than that of the hard-output detector. This paper presents the first soft-output message-passing detector (MPD), which is also integrated with a high-throughput polar decoder. The algorithm and architecture are designed concurrently to improve the hardware performance. The proposed techniques, including the adaptive variance estimation and reliable symbol detection, reduce the complexity of the MPD by 98.3%, and enable soft output for the outer polar decoder. Compared to the state-of-the-art hard-output design, the proposed MPD achieves a 5.86× higher throughput-to-area ratio (TAR) with 54.3% lower energy dissipation, despite the soft output. The proposed bidirectional-propagation belief propagation decoder is devised to reduce the critical path and to increase the throughput. The proposed polar decoder can improve the TAR by 35% with a comparable area and energy compared with the state-of-the-art polar decoder. A polar-coded massive MIMO receiver that supports a length-1024 rate-1/2 polar code, 128 receive antennas, and eight users is designed and implemented, and delivers a throughput of 7.61 Gb/s.