They are most likely not reinfections. You can still have the virus reproducing in your body, yet have recovered and are no longer spreading it. If you’re in this post-infection period, you’ll still intermittently test positive (especially when your stool is sampled). This is similar to EBV or HIV.
These tests (RT-PCR) also aren’t sensitive enough to test whether the virus is alive or dead, they test for the presence of genetic material.
Viruses are not living things.
How many cases? Are you talking about the woman in Japan?