  Anomaly analysis has been an important and powerful tool in studying
nonperturbative physics for decades. The anomaly inflow mechanism provides an
intuitive interpretation of the bulk-boundary correspondence in topological
systems. In this work, we study global anomalies in systems with
symmetry-preserving Luttinger surfaces, i.e. the manifolds of fermionic Green's
function zeros in the momentum space at zero energy, described by nonlocal
effective theories. We view the nonlocal effective theories as a result of
integrating out some low energy states. Assuming that the states integrated out
do not make extra contributions to the anomalies, we discuss the simplest
Lagrangian describing a gapless Dirac zero and a two-pole variant, their global
anomalies, and the bulk-boundary correspondence. We then consider the
constraints on possible phases with Green's function zeros of Dirac type, such
as non-Fermi liquids and emergent gapless quasiparticles on Luttinger surfaces.
We also provide some perspectives on why the nonlocal fermionic effective
theory discussed by Golterman and Shamir (arXiv: 2311.12790) is not a suitable
starting point for a symmetrically gapped phase.
