  QCD at nonzero baryon chemical potential suffers from the sign problem, due
to the complex quark determinant. Complex Langevin dynamics can provide a
solution, provided certain conditions are met. One of these conditions,
holomorphicity of the Langevin drift, is absent in QCD since zeroes of the
determinant result in a meromorphic drift. We first derive how poles in the
drift affect the formal justification of the approach and then explore the
various possibilities in simple models. The lessons from these are subsequently
applied to both heavy dense QCD and full QCD, and we find that the results
obtained show a consistent picture. We conclude that with careful monitoring,
the method can be justified a posteriori, even in the presence of
meromorphicity.
