Importance of Phase in FFT of an image

While processing digital images in the Fourier domain, mostly we exploit the amplitude and not the phase. This could be because the amplitude is much more structured and the amplitude spectrum reveals a lot about the spatial domain as illustrated in HIPR and other answers.

I am really curious to know what does the phase encapsulates, what can we infer from it, what kinds of operations can we perform on the phase of FFT of an image to achieve some objectives?

My specific interest lies in fusing two images in the frequency domain, but any PDF or well-documented links on understanding the phase of FFT(image) will help.

------ Edit (following images as suggested by TimWescott in comments)    • The phase contains half of all information in the image. Do a mental experiment: multiplying your whole image's FFT with $\sqrt{-1}$ does nothing to its amplitude, but what happens to the image. Now, imagine multiplying each element of the FFT with a random complex factor of magnitude 1 (i.e. not changing the amplitude, but the phase), thereby completely removing the original phase information. What's left of your image? Jan 18 '21 at 15:36
• Better than a mental experiment: grab an image, and get it into your image processing tool. Take it's 2D FFT. Now calculate $x_{mag} = |x|$, where $|\cdot|$ is the magnitude of each FFT bin. Then calculate $x_\theta = x / x_{mag}$ (again, point-by-point). Then display the IFFT of $x_{mag}$ and $x_\theta$ as images. See what you think. Jan 18 '21 at 18:42
• @TimWescott I have added the images suggested by you in the question, but still cannot understand what is that phase is encapsulating other that without it cannot recover the original image. Jan 22 '21 at 13:18 