  We construct a Wannier basis for twisted bilayer graphene that is projected
only from the Bloch functions of the twisted bilayer flat bands. The $C_3$ and
$C_{2} \mathcal{T}$ symmetries act locally on the Wannier functions while the
Wannier function charge density is strongly peaked at the triangular sites and
becomes fully sublattice-polarized in the chiral limit. The Wannier functions
have a power-law tail, due to the topological obstruction, but most of the
charge density is concentrated within one unit cell so that the on-site local
Coulomb interaction is much larger than the further neighbor interactions and
in general the Hamiltonian parameters may be accurately estimated from a modest
number of Wannier functions. One exception is the momentum space components of
the single-particle Hamiltonian, where because of the topological obstruction
convergence is non-uniform across the Brillouin zone. We observe, however, that
mixed position and momentum space representations may be used to avoid this
difficulty in the context of quantum embedding methods. Our work provides a new
route to study systems with topological obstructions and paves the way for the
future investigation of correlated states in twisted bilayer graphene,
including studies of non-integer fillings and temperature dependence.
