I was at the grocery store today and noticed they sold root beer extract, so ... here's the recipe.
Changes made from that recipe:
I used 1/4 cup brown sugar, 1/4 cup white sugar, and 1/4 cup honey. In retrospect I should have added the honey after step 9 ("Half fill the bottle with fresh cool tap water") because it appears to have trapped some of the yeast against the bottom of the bottle. I'm hoping it will dissolve as the mixture warms up. Edit:
Actually just jostling the bottle around for 10-15 minutes seem to have done the trick -- though there is still a chunk of undissolved brown sugar at the moment.
In addition to the ingredients listed, I added 1/8 tsp cinnamon, 1/16 tsp nutmeg, a splash of vanilla, and a little lemon zest. Any or all of these might be mistakes but I gotta be me.
I tried using a little less water than I did with the ginger beer but I'm not sure if I succeeded or not -- if so it is probably not enough of a difference to matter. Oh well.
The claim is that this will take 3-4 days to be done, which I am a bit doubtful about after the ginger ale was done about three times as fast as it was supposed to be. We'll see how it goes.
Since last time I tried making soda I have invested in a funnel and a sieve, which should help to facilitate further experiments along these lines.