Cross-correlation - Why is there a complex part after FFT/IFFT when deleting the mirrored part of the spectrum?
If the input signal is real-valued, then the DFT returns a sequence of length $N$ that satisfies $$X[N-n]=X^*[n]$$

As one usually plots the magnitude of the spectrum, then the relationship of interest is:

$$|X[N-n]|=|X^*[n]|\implies|X[N-n]|=|X[n]|$$

When you work with real-valued signals, one tends to "ignore" the second half of the FFT spectrum due to this symmetry property.

However, if the signal is complex, then both sides of the spectrum are fundamental, as the symmetry is not accomplished anymore. When you call the function ifft after discarding one side of the spectrum, the function is assuming that the original FFT (the input) is non-symmetrical (as you deleted one half of it), so the time-domain sequence from which it came from must be complex. That's why you get a complex-valued signal when coming back to the time-domain.