  The real Chern insulator state, featuring nontrivial real Chern number and
second-order boundary modes, has been revealed in a few two-dimensional
systems. The concept can be extended to three dimensions (3D), but a proper
material realization is still lacking. Here, based on first-principles
calculations and theoretical analysis, we identify the recently synthesized
bulk ${\gamma}$-graphyne as a 3D real Chern insulator. Its nontrivial bulk
topology leads to topological hinge modes spreading across the 1D edge
Brillouin zone. Under compression of the interlayer distance, the system can
undergo a topological phase transition into a real nodal-line semimetal, which
hosts three bulk nodal rings and topological boundary modes on both surfaces
and hinges. We also develop a minimal model which captures essential physics of
the system.
