Physical unclonable functions (PUFs) leveraging minute silicon process variations have been proposed for generating the device-tied secret keys. When creating the PUF-based keys, the high energy and area costs for implementing error corrections can far exceed the costs of the basic PUF circuits alone. It is critical to minimize the end-to-end cost of the reliable key generation; otherwise, the application of PUFs would be greatly limited. In this paper, we propose a new PUF scheme that uses a circuit with two different oscillation modes (bimodal oscillation) to realize the autonomous majority voting and improve PUF reliability. This design is novel and efficient because the voting procedure is realized with a self-timed oscillating circuit without orchestration by a clock signal. We use both circuit simulation and FPGA implementation to evaluate the reliability tradeoffs achieved by different parameterizations of the design, to show that the design performs well across a range of supply voltages, and to quantify the robustness of the design across a broad range of operating temperatures.