Used in resistive superconducting fault current limiters (R-SFCLs), REBCO tapes are required to have both large current limiting resistance and high over-current bearing capacity. The contradiction between high resistance and large over-current is the main difficulty in the design of REBCO-tape structure for R-SFCLs. In this paper, an asymmetric stabilizer layer structure of REBCO tapes has been proposed, fabricated and tested for R-SFCLs. Comparing to traditional tapes with a symmetric stabilizer structure, the REBCO tape with new structure showed a 59 % improvement in over-current bearing capacity: the peak current increased from 1020 A to 1620 A while keeping the same total resistance. Simulation calculation on the current distributions in every layer showed that the asymmetric structure is superior to the symmetric structure.