More precisely, let's say I apply a 45 degrees rotation to an image (in the spatial domain) say, in Matlab :

Ir=imrotate(myImage,45,'crop');
FT_I=fft2(I);

In the magnitude, i.e. abs(FT_I) is it clear that the spectrum has be rotated too.

For example, trying to reconstruct/sort of "unrotate", wrt center (with shifted DC in center), in the frequency domaine like:

FT_Is=fftshift(FT_I);
I_rec_FT=abs(ifftshift(imrotate(FT_Is,45,'crop'))).*exp(-i*angle(ifftshift(imrotate(FT_Is,-45,'crop'))));

...does not make sense (ifft2()) does not show an image that seems at all sort of unrotated.

Therefore, what exactly happens with the phase part?

"unrotated" unsuccessfully, image.