  Topologically protected fermionic quasiparticles occur in metals with band
degeneracy as a consequence of band structure topology. Here we unveil
topological semimetal and metal phases in a variety of non-symmorphic collinear
antiferromagnets with glide reflection symmetry, a combination of mirror and
half-lattice translation. We find gapless phases with Dirac points having
multiple symmetry-protection as well as electronic structures with triple and
quadruple band-crossing points. Glide semimetal is shown to be converted into a
topological phase with non-trivial $\mathbb{Z}_2$ topological charges at the
Dirac points due to inversion and time-inversion symmetry combination. More
striking is the emergence of a hidden non-unitary relation between the states
in the glide sectors that provide a general mechanism to get multiple band
touching points. The split Fermi points uncover a $\mathbb{Z}_2$ protection
that drives the changeover of the multiple-degenerate gapless phase into a
topological metal built from their connection through distinct Fermi lines.
Besides a new perspective of ordered states in complex materials, our findings
indicate that novel topological gapless phases and edge states may occur in a
wide class of magnetic systems.
