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We propose a new probe of inflationary gravitational waves (IGWs): thecross-correlation of the lensing of inflationary $B$-mode polarization with alarge-scale structure (LSS) tracer, which can also be a cosmic microwavebackground (CMB) lensing map. This is equivalent to measuring a three-pointfunction of two CMB $B$-modes and an LSS tracer. We forecast expected$1\,\sigma$ constraints on the tensor-to-scalar ratio $r$, albeit with asimplistic foreground treatment, and find constraints of $\sigma_r \simeq 7\times 10^{-3}$ from the correlation of CMB-S4-Deep $B$-mode lensing and LSSTgalaxies, $\sigma_r \simeq 5 \times 10^{-3}$ from the correlation ofCMB-S4-Deep $B$-mode lensing and CMB-S4-Deep CMB lensing, and $\sigma_r \simeq10^{-2}$ from the correlation of LiteBIRD $B$-mode lensing and CMB-S4-Widelensing. Because this probe is inherently non-Gaussian, simple Gaussianforegrounds will not produce any biases to the measurement of $r$. While adetailed investigation of non-Gaussian foreground contamination for differentcross-correlations will be essential, this observable has the potential to be auseful probe of IGWs, which, due to different sensitivity to many potentialsources of systematic errors, can be complementary to standard methods forconstraining $r$.

New probe of inflationary gravitational waves: cross-correlations of lensed primary CMB B-modes with large-scale structure