  Correlated insulators and superconductivity have been observed in
"magic-angle" twisted bilayer graphene, when the nearly flat bands close to
neutrality are partially filled. While a momentum-space continuum model
accurately describes these flat bands, interaction effects are more
conveniently incorporated in tight-binding models. We have previously shown
that no fully symmetric tight-binding model can be minimal, in the sense of
capturing just the flat bands, so extended models are unavoidable. Here, we
introduce a family of tight-binding models that capture the flat bands while
simultaneously retaining all symmetries. In particular, we construct three
concrete models with five, six, or ten bands per valley and per spin. These
models are also faithful, in that the additional degrees of freedom represent
energy bands further away from neutrality, and they serve as optimal starting
points for a controlled study of interaction effects. Furthermore, our
construction demonstrates the "fragile topology" of the nearly flat bands;
i.e., the obstruction to constructing exponentially localized Wannier functions
can be resolved when a particular set of trivial bands is added to the model.
