  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.

Astrophysics of Galaxies;Solar and Stellar Astrophysics