  Upcoming cosmic microwave background (CMB) data can be used to explore
harmonic 3-point functions that involve the B-mode component of the CMB
polarization signal. We focus on bispectra describing the non-Gaussian
correlation of the B-mode field and the CMB temperature anisotropies (T) and/or
E-mode polarization, i.e. <TTB>, <EEB>, and <TEB>. Such bispectra probe
violations of the tensor consistency relation: the model-independent behavior
of cosmological correlation functions that involve a large-wavelength tensor
mode (gravitational wave). An observed violation of the tensor consistency
relation would exclude a large number of inflation models. We describe a
generalization of the Komatsu-Spergel-Wandelt (KSW) bispectrum estimator that
allows statistical inference on this type of primordial non-Gaussianity with
data of the CMB temperature and polarization anisotropies. The generalized
estimator shares its statistical properties with the existing KSW estimator and
retains the favorable numerical scaling with angular resolution. In this paper
we derive the estimator and present a set of Fisher forecasts. We show how the
forecasts scale with various experimental parameters such as lower and upper
angular band-limit, relevant for e.g. the upcoming ground-based Simons
Observatory experiment and proposed LiteBIRD satellite experiment. We comment
on possible contaminants due to secondary cosmological and astrophysical
sources.
