Hi Peter,
The color space is a conversion table, calculates the output image color values for applicable to the next device. Without such conversion, the saturated part of the colors will simple loose the difference of some shades. ( That is why, there are 4 type of compression method https://helpx.adobe.com/acrobat/using/color-settings.html ) The ProPhoto color space allow almost full 16 bit resolution for the R & G & B colors. In the real life the output devices unable to reproduce all. To keep the shades on different output devices, the chosen color space will ensure to get the closest quality to the original.
You can make a test by own. In PS make 3 rectangle, fill with gradient of RGB colors. Export them in TIFF by 3 different color spaces. Open all 3 in DPL and you will see the difference.
In practice, the best monitors has 96% of Adobe RGB color fidelity, the best printers - having 16 bit driver, which is exceptional - never able to reproduce more than 90% of Adobe RGB. I never heard about any monitor or printer able to reproduce more than Adobe RGB and get closer to ProPhoto color space.
Endre