Several recent spectroscopic investigations have presented conflicting results on the existence of Na-rich asymptotic giant branch (AGB) stars in the Galactic globular cluster M 4 (NGC 6121). The studies disagree on whether or not Na-rich red giant branch (RGB) stars evolve to the AGB. For a sample of previously published HERMES/AAT AGB and RGB stellar spectra, we present a re-analysis of O, Na, and Fe abundances, and a new analysis of Mg and Al abundances; we also present CN band strengths for this sample, derived from low-resolution AAOmega spectra. Following a detailed literature comparison, we find that the AGB samples of all studies consistently show lower abundances of Na and Al, and are weaker in CN, than RGB stars in the cluster. This is similar to recent observations of AGB stars in NGC 6752 and M 62. In an attempt to explain this result, we present new theoretical stellar evolutionary models for M 4; however, these predict that all stars, including Na-rich RGB stars, evolve on to the AGB. We test the robustness of our abundance results using a variety of atmospheric models and spectroscopic methods; however, we do not find evidence that systematic modelling uncertainties can explain the apparent lack of Na-rich AGB stars in M 4. We conclude that an unexplained, but robust, discordance between observations and theory remains for the AGB stars in M 4.