We present measurements of the electron helicity asymmetry in quasi-elastic proton knockout from 2H and 12C 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 , with a polar angle coverage up to about 8°. The missing energy resolution enabled us to determine the asymmetries for knock-out resulting in different states of the residual 11B system. We find that the helicity asymmetry for p-shell knockout from 12C depends on the final state of the residual system and is relatively large (up to ≈0.16), especially at low missing momentum. It is considerably smaller (up to ≈0.01) for s-shell knockout from both 12C and 2H. The data for 2H are in very good agreement with theoretical calculations, while the predictions for 12C exhibit differences with respect to the data.