  Weyl fermions are hypothetical chiral particles that can also manifest as
excitations near three-dimensional band crossing points in lattice systems.
These quasiparticles are subject to the Nielsen-Ninomiya "no-go" theorem when
placed on a lattice, requiring the total chirality across the Brillouin zone to
vanish. This constraint results from the topology of the (orientable) manifold
on which they exist. Here, we ask to what extent the concepts of topology and
chirality of Weyl points remain well-defined when the underlying manifold is
non-orientable. We show that the usual notion of chirality becomes ambiguous in
this setting, allowing for systems with a non-zero total chirality. This
circumvention of the Nielsen-Ninomiya theorem stems from a generic
discontinuity of the vector field whose zeros are Weyl points. Furthermore, we
discover that Weyl points on non-orientable manifolds carry an additional
$\mathbb{Z}_2$ topological invariant which satisfies a different no-go theorem.
We implement such Weyl points by imposing a non-symmorphic symmetry in the
momentum space of lattice models. Finally, we experimentally realize all
aspects of their phenomenology in a photonic platform with synthetic momenta.
Our work highlights the subtle but crucial interplay between the topology of
quasiparticles and of their underlying manifold.
