We derive a multidimensional instanton theory for calculating ground-state tunneling splittings in Cartesian coordinates for general paths. It is an extension of the method by Mil'nikov and Nakamura [J. Chem. Phys. 115, 6881 (2001)] to include asymmetric paths that are necessary for calculating tunneling splitting patterns in multi-well systems, such as water clusters. The approach avoids multiple expensive matrix diagonalizations to converge the fluctuation prefactor in the ring-polymer instanton (RPI) method, and instead replaces them by an integration of a Riccati differential equation. When combined with the string method for locating instantons, we avoid the need to converge the calculation with respect to the imaginary time period of the semiclassical orbit, thereby reducing the number of convergence parameters of the optimized object to just one: the number of equally spaced system replicas used to represent the instanton path. The entirety of the numerical effort is thus concentrated in optimizing the shape of the path and evaluating hessians along the path, which is a dramatic improvement over RPI. In addition to the standard instanton approximations, we neglect the coupling of vibrational modes to external rotations. The method is tested on the model potential of malonaldehyde and on the water dimer and trimer, giving close agreement with RPI at a much-reduced cost. Published under license by AIP Publishing.