This is a little different than "Ginger Beer", so I'm not sure what you are exactly making because your post does not make it completely clear. However, I have had success making a naturally fermented Ginger Soda from a recipe our of Sandor Katz's wonderful book, "Wild Fermentation" which I recommend to any beer maker as an interesting read.
Anyway, essentially the recipe starts with making what he calls a Ginger Bug. This is the "yeast" of the process. Essentially, you mix 1 cup water with 2 teaspoons fresh grated ginger and 2 teaspoons white sugar. Leave that covered with a cloth on your counter for 2 days to a week.. until it starts bubbling. Add more ginger and sugar every 1-2 days as needed.
(This recipe makes 1 gallon, so adjust accordingly.)
Now, when you go to make your soda.. bring 1 gallon water to a boil with 1.5 cups white sugar and 2-6 inches grated fresh ginger (depending on how strong you want the ginger taste to be). Let that simmer for 15-20 minutes.
As it cools, you can optionally add the juice of 1-2 lemons for another flavor. In my latest batch, I added carrot juice for more fun!
Once it cools down to 80F or so, add the strained ginger bug from above. Katz says to bottle it directly in your bottles, but I've had that be over-carbonated, so I ferment mine in a small carboy for about 10 days. Then I bottle with priming sugar to get the right level of carbonation.
Anyway, that's a naturally fermented ginger soda. Super tasty!