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In the glossary of the AMS book on Quantum Fields and Strings it is stated that

An anomalous theory does not make sense quantum mechanically, so anomaly cancellation is a fundamental requirement of quantum theories.

However, from what I understand, a chiral anomaly means that quantum fluctuations destroy the axial symmetry of the classical theory, whose reality is an observed fact.

Are these two different kinds of anomalies?

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A theory with an anomalous global symmetry, like chiral symmetry, is not problematic. Global symmetries are physical, and we can break them. Breaking gauge symmetry is a problem, because gauge symmetry is just a redundancy in labelling things. A $d$-dimensional theory with an anomalous gauge symmetry is telling us that it can't exist in just $d$ dimensions, but it could exist as the boundary of a $d+1$ dimensional theory. (See Wen, arXiv:1303.1803)

Some more details on anomalies in global vs. gauge symmetries are in this answer by Luboš Motl.

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