Afraid I can't help much because I don't use Affinity. But I downloaded your file. This is what I saw.
1. I get the same strange display in my Fastsone (V6.2).
2. Adobe Bridge CC and Lightroom display the file OK. So does Windows Explorer.
3. When I open the file in Photoshop I get a 16bit sRGB image with LZW compression. Although it has only one layer, it shows as "Layer 0", which means Photoshop thinks it is not a "flat" file.
4. When I save the file from Photoshop, the result looks OK in Faststone. Regardless of what compression or bit depth I use. And regardless whether it is truly flat or not. However, your original is only 866kb. The file saved by Photoshop is 1800kb to 1900kb (depending on compression).
The TIF file format is a complicated beast. It's really a "container" specification, and can contain many different types of objects - bitmaps, jpegs, text, etc. Not all programs follow the specs properly. In your case, it's hard to tell if Affinity or Faststone is the bad actor. I'd bet they each would blame the other guy.
One note - don't use LZW compression on 16bit files. There is a well known and old "bug" with LZW compression that causes it to create 16bit files that are actually larger than if no compression is used at all. LZW works great on 8bit files, and is a lot faster than ZIP. I'm curious what happens if you save your file from Affinity in 16bit and no compression. Do you still get the strange display?
Another note - Faststone has trouble with some TIF files created by Photoshop too. If a Photoshop TIF has extra alpha channels or paths, Faststone gives an incorrect display.