  Integrating over a continuum Majorana fermion formally yields a functional
pfaffian. We show that the phase of this pfaffian is ambiguous, as it depends
on the choice of basis. This ambiguity is naturally resolved within a
non-perturbative lattice definition, allowing us to discuss the relation
between the phase of the lattice pfaffian and the effective $\theta$ angle of
the theory. We also resolve an apparent paradox regarding the induced $\theta$
angle when a theory of $N$ Dirac fermions in a real representation of the gauge
group is re-expressed in terms of $2N$ Majorana fermions. We discuss how all
this is reflected in chiral perturbation theory.
