  We present measurements of the electron helicity asymmetry in quasi-elastic
proton knockout from $^{2}$H and $^{12}$C nuclei by polarized electrons. This
asymmetry depends on the fifth structure function, is antisymmetric with
respect to the scattering plane, and vanishes in the absence of final-state
interactions, and thus it provides a sensitive tool for their study. Our
kinematics cover the full range in off-coplanarity angle $\phi_{pq}$, with a
polar angle $\theta_{pq}$ coverage up to about 8 degrees. The missing energy
resolution enabled us to determine the asymmetries for knock-out resulting in
different states of the residual $^{11}$B system. We find that the helicity
asymmetry for $p$-shell knockout from $^{12}$C depends on the final state of
the residual system and is relatively large (up to $\approx 0.16$), especially
at low missing momentum. It is considerably smaller (up to $\approx 0.01$) for
$s$-shell knockout from both $^{12}$C and $^2$H. The data for $^2$H are in very
good agreement with theoretical calculations, while the predictions for
$^{12}$C exhibit differences with respect to the data.
