We address the real-time tracking problem of a partially observable Markov source under sampling and transmission costs in an energy harvesting system with an unreliable communication channel. We provide a semantic-aware optimal sampling and transmission policy that minimizes the average value of a general distortion subject to an energy causality constraint. We formulate a partially observable Markov decision process (POMDP) problem. To solve the problem, we cast it into a belief MDP problem. Subsequently, by effectively bounding the belief space, we formulate a finite-state MDP problem, which is solved using relative value iteration. The simulation results demonstrate the effectiveness of the derived policy and highlight the significant impact of the source dynamics on performance.