  Real topological phases protected by the spacetime inversion (P T) symmetry
are a current research focus. The basis is that the P T symmetry endows a real
structure in momentum space, which leads to Z2 topological classifications in
1D and 2D. Here, we provide solutions to two outstanding problems in the
diagnosis of real topology. First, based on the stable equivalence in K-theory,
we clarify that the 2D topological invariant remains well defined in the
presence of nontrivial 1D invariant, and we develop a general numerical
approach for its evaluation, which was hitherto unavailable. Second, under the
unit-cell convention, noncentered P T symmetries assume momentum dependence,
which violates the presumption in previous methods for computing the
topological invariants. We clarify the classifications for this case and
formulate the invariants by introducing a twisted Wilson-loop operator for both
1D and 2D. A simple model on a rectangular lattice is constructed to
demonstrate our theory, which can be readily realized using artificial
crystals.
