  Mixing of axion fields is widely used to generate EFTs with
phenomenologically advantageous features, such as hierarchies between axion
couplings to different gauge fields and/or large effective field ranges. While
these features are strongly constrained by periodicity for models with only a
single axion, mixing has been used in the literature (sometimes incorrectly) to
try to evade some of these constraints. In this paper, we ask whether it is
possible to use axion mixing to generate an EFT of axions that evades these
constraints by flowing to a theory of a non-compact scalar in the IR. We
conclude that as long as the light axion is exactly massless, it will inherit
the periodicity and associated constraints of the UV theory. However, by giving
the light axion a mass, we can relax these constraints with effects
proportional to the axion mass squared, including non-quantized couplings and
the realignment of monodromy to a light axion with a larger field range. To
show this, we consider various examples of axions mixing with other axions or
with non-compact scalar fields, and work in a basis where coupling quantization
is manifest. This basis makes it clear that in the case where an axion is eaten
through the Higgs or Stuckelberg mechanism, the light axion does not have a
large effective field range, in contrast to some recent claims in the
literature. Additionally, we relate our results about axion EFTs to a
well-known fact about gauge theory: that QFTs with compact gauge groups in the
UV flow to QFTs with compact gauge groups in the IR, and make this
correspondence precise in the 2+1 dimensional case.
