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Stochastic reduced-order modeling based on time-dependent bases (TDBs) has proven successful for extracting and exploiting low-dimensional manifold from stochastic partial differential equations (SPDEs). The nominal computational cost of solving a rank-$r$ reduced-order model (ROM) based on time-dependent basis, a.k.a. TDB-ROM, is roughly equal to that of solving the full-order model for $r$ random samples. As of now, this nominal performance can only be achieved for linear or quadratic SPDEs -- at the expense of a highly intrusive process. On the other hand, for problems with non-polynomial nonlinearity, the computational cost of solving the TDB evolution equations is the same as solving the full-order model. In this work, we present an adaptive sparse interpolation algorithm that enables stochastic TDB-ROMs to achieve nominal computational cost for generic nonlinear SPDEs. Our algorithm constructs a low-rank approximation for the right hand side of the SPDE using the discrete empirical interpolation method (DEIM). The presented algorithm does not require any offline computation and as a result the low-rank approximation can adapt to any transient changes of the dynamics on the fly. We also propose a rank-adaptive strategy to control the error of the sparse interpolation. Our algorithm achieves computational speedup by adaptive sampling of the state and random spaces. We illustrate the efficiency of our approach for two test cases: (1) one-dimensional stochastic Burgers' equation, and (2) two-dimensional compressible Navier-Stokes equations subject to one-hundred-dimensional random perturbations. In all cases, the presented algorithm results in orders of magnitude reduction in the computational cost.