  The study of higher-order and real topological states as well as the material
realization have become a research forefront of topological condensed matter
physics in recent years. Twisted bilayer graphene (tbG) is proved to have
higher-order and real topology. However whether this conclusion can be extended
to other two-dimensional twisted bilayer carbon materials and the mechanism
behind it lack explorations. In this paper, we identify the twisted bilayer
$\alpha$-graphyne (tbGPY) at large twisting angle as a real Chern insulator
(also known as Stiefel-Whitney insulator) and a second-order topological
insulator. Our first-principles calculations suggest that the tbGPY at
21.78$^\circ$ is stable at 100 K with a larger bulk gap than the tbG. The
non-trivial topological indicators, including the real Chern number and a
fractional charge, and the localized in-gap corner states are demonstrated from
first-principles and tight-binding calculations. Moreover, with $\mathcal
C_{6z}$ symmetry, we prove the equivalence between the two indicators, and
explain the existence of the corner states. To decipher the real and
higher-order topology inherited from the Moir\'e heterostructure, we construct
an effective four band tight-binding model capturing the topology and
dispersion of the tbGPY at large twisting angle. A topological phase transition
to a trivial insulator is demonstrated by breaking the $\mathcal C_{2y}$
symmetry of the effective model, which gives insights on the trivialization of
the tbGPY as reducing the twisting angle to 9.43$^\circ$ suggested by our
first-principles calculations.
