The problem of efficient ultra-massive multipleinput multiple-output (UM-MIMO) data detection in terahertz (THz)-band non-orthogonal multiple access (NOMA) systems is considered. We argue that the most common THz NOMA configuration is power-domain superposition coding over quasioptical doubly-massive MIMO channels. We propose spatial tuning techniques that modify antenna subarray arrangements to enhance channel conditions. Towards recovering the superposed data at the receiver side, we propose a family of data detectors based on low-complexity channel matrix puncturing, in which higher-order detectors are dynamically formed from lower-order component detectors. The proposed solutions are first detailed for the case of superposition coding of multiple streams in pointto-point THz MIMO links. Then, the study is extended to multiuser NOMA, in which randomly distributed users get grouped into narrow cell sectors and are allocated different power levels depending on their proximity to the base station. Successive interference cancellation is shown to be carried with minimal performance and complexity costs under spatial tuning. Approximate bit error rate (BER) equations are derived, and an architectural design is proposed to illustrate complexity reductions. Under typical THz conditions, channel puncturing introduces more than an order of magnitude reduction in BER at high signal-to-noise ratios while reducing complexity by approximately 90%.