  Extragalactic foregrounds are known to constitute a limiting systematic in
temperature-based CMB lensing with AdvACT, SPT-3G, Simons Observatory and CMB
S4. Furthermore, since these foregrounds are emitted at cosmological distances,
they are also themselves lensed. The correlation between this foreground
lensing and CMB lensing causes an additional bias in CMB lensing estimators. In
this paper, we quantify for the first time this "lensed foreground bias" for
the standard CMB lensing quadratic estimator, the CMB shear and the CMB
magnification estimators, in the case of Simons Observatory and in the absence
of multi-frequency component separation. This percent-level bias is highly
significant in cross-correlation of CMB lensing with LSST galaxies, and
comparable to the statistical uncertainty in CMB lensing auto-spectrum. We
discuss various mitigation strategies, and show that "lensed foreground
bias-hardening" methods can reduce this bias at some cost in signal-to-noise.
The code used to generate our theory curves is publicly available at
https://github.com/EmmanuelSchaan/LensedForegroundBias .
