The fundamental problem of our interest here is soft-input soft-output multiple-input multiple-output (MIMO) detection. We propose a method, referred to as subspace marginalization with interference suppression (SUMIS), that yields unprecedented performance at low and fixed (deterministic) complexity. Our method provides a well-defined tradeoff between computational complexity and performance. Apart from an initial sorting step consisting of selecting channel-matrix columns, the algorithm involves no searching nor algorithmic branching; hence the algorithm has a completely predictable run-time and allows for a highly parallel implementation. We numerically assess the performance of SUMIS in different practical settings: full/partial channel state information, sequential/iterative decoding, and low/high rate outer codes. We also comment on how the SUMIS method performs in systems with a large number of transmit antennas.