This paper proposes a wide-area backup protection (WABP) method for transmission systems using sparse synchronized/unsynchronized PMU measurements. The method is aimed at addressing practical challenges such as temporary loss of the time-synchronization signal (LTSS), sparse PMU coverage, and communication failures and latencies. A linear and computationally efficient formulation is proposed to identify the faulted line in near real-time based on the superimposed-circuit concept. An index is proposed that quantifies the mismatch degrees between the expected and observed superimposed phasors without requiring full network observability. The method can work well with unsynchronized measurements without imposing a significant computational burden. This is achieved by canceling out the effect of angle drifts caused by LTSS from the equations. Since no matrix inversion is involved, sparse PMU measurements do not result in singularity, and thus, the unsolvability of the equations. A technique is proposed to assess the feasibility of faulted-line identification by a given set of PMUs. Being robust against measurement and parameter errors, the method performs well with PMUs of different reporting rates regardless of the fault distance, type, and resistance. More than 200,000 simulations conducted on the IEEE 39-bus test system verify the effectiveness of the proposed WABP method.