We consider the application of asymmetric fullduplex (AFD) communications in wireless local area networks (WLANs), exemplified by a Wi-Fi system. A full-duplex (FD)enabled Wi-Fi access point communicates simultaneously uplink (UL) and downlink (DL) with a pair of half-duplex (HD) Wi-Fi stations (STAs). AFD enhances spectrum efficiency and reduces latency for STAs; however, it faces challenges related to node selection, switching between AFD and HD modes, and rate control. In this paper, we propose the asymmetric FD-mode and rate adaptation (AFRA) scheme. AFRA relies on partially observable Markov decision process (POMDP) to adapt the UL and DL transmission rates as well as the transmission mode (i.e., AFD or HD; UL-only or DL-only), considering the future expected channel and interference conditions. Our simulation results indicate that AFRA outperforms classical rate-adaptation schemes and achieves up to 95% of the optimal performance of AFD communications.