What would the behavior be of an FFT without the phase (imaginary part)
The real part isn't the amplitude.  

The imaginary part is the sine wave part of the FFT.  Thus exactly periodic- in-aperture sine waves (with a phase of zero at the window start) won't show up on the real part, and the FFT real part will see a zero signal.  Only sinusoids with a periodic cosine wave (with a phase of zero at the window start) component will show up at all.

The "beating" you are seeing is your signal alternating between being in and out of phase compared to a cosine basis vector.