What actually happens to reduce the perceived color in a 'white rainbow" or "fog-bow"?
The reason why fog bows lack colour, compared to rainbows, is due to the size of the drops of water.

Fog is composed of very small drops of water - less than 0.05 mm diameter. Because of this the wavelength of light is critical, with diffraction smearing out colours that the larger drops of water in a rainbow would make.