I think with the full FTs you have to take the real part, the Friedel mates have equal and opposite phase differences that will cancel out so the result should be real, but numerical error will add a small imaginary component.
BTW, it’s common in x-ray crystallography classes to prove that the electron density is real. You should be able to do it if you think about the density equation:
Hint - expand the exponent using Euler’s formula.
Here are two slightly different approaches:
https://www-structmed.cimr.cam.ac.uk/Course/Fourier/Fourier.html
PS the Coulomb potential is real for the same reason, this time there’s no difference 