Authentication is a fundamental requirement and a crucial topic in wireless communications. This paper focuses on improving the security of the prior tag-based Physical-Layer Authentication (PLA) schemes. We propose two phase-tag-based PLA schemes to overcome the limitations of prior tag-based PLA schemes. First, we propose the Phase-Tag-based PLA (PT-PLA) scheme, which superimposes a tag to the phase of the transmitted signal rather than the amplitude. Second, we propose the Adaptive Phase-Tag-based PLA (APT-PLA) scheme, where an appropriate parameter of the PT-PLA scheme is adaptively set for achieving a better trade-off among robustness, security, and compatibility. Rigorous theoretical analyses of the proposed schemes are conducted in high-order modulation systems, such as$M$-PSK and${M}$-QAM systems, with a focus on evaluating their robustness, security, and compatibility. We also discuss advantages and disadvantages of these schemes and offer helpful recommendations according to different scenarios. Furthermore, we implement the proposed schemes and perform extensive simulations to compare their performance comprehensively. Simulation results demonstrate a perfect match with the theoretical results, verifying the superiority of the proposed schemes over the prior schemes.