We report the peculiar chemical abundance patterns of eleven atypical Milky Way (MW) field red giant stars observed by the Apache Point Observatory Galactic Evolution Experiment (APOGEE). These atypical giants exhibit strong Al and N enhancements accompanied by C and Mg depletions, strikingly similar to those observed in the so-called second-generation (SG) stars of globular clusters (GCs). Remarkably, we find low-Mg abundances ([Mg/Fe]$<$0.0) together with strong Al and N overabundances in the majority (5/7) of the metal-rich ([Fe/H]$gtrsim - 1.0$) sample stars, which is at odds with actual observations of SG stars in Galactic CGs of similar metallicities. This chemical pattern is unique and unprecedented among MW stars, posing urgent questions about its origin. These atypical stars could be former SG stars of dissolved GCs formed with intrinsically lower abundances of Mg and enriched Al (subsequently self-polluted by massive AGB stars) or the result of exotic binary systems. We speculate that the stars Mg-deficiency as well as the orbital properties suggest that they could have an extragalactic origin. This discovery should guide future dedicated spectroscopic searches of atypical stellar chemical patterns in our Galaxy; a fundamental step forward to understand the Galactic formation and evolution.