BPS decoupling limits of string and M-theory have led to profound insights into gravity and non-perturbative quantum theory, most notably with the AdS/CFT correspondence. A large class of such limits are related by U-duality to discrete lightcone quantization limits (DLCQs). However, there are natural BPS decoupling limits within the AdS/CFT correspondence that cannot be seen as DLCQ limits. Therefore, we introduce a new kind of BPS decoupling limit in which the relevant BPS states carry momentum along circle fibers geometrically twisted over a base space, unlike for a DLCQ limit. In the AdS/CFT correspondence these limits are associated with BPS states of the global symmetry algebra  PSU(2,2|4). In general, string fluctuations in a twisted BPS decoupling limit are described by non-relativistic sigma-models on emergent non-Lorentzian target spaces with a natural symplectic structure. This  further supports the broader connection between BPS limits and non-Lorentzian geometry. We introduce analogous limits of D-branes as well, probing the same symplectic structure as the string. For AdS/CFT, these results point to corners with tractable and explicit quantum-mechanical descriptions of non-perturbative regimes.