In a medium bowl, mash the fruit, peels, sugar, and water and store this in a jar covered with cheesecloth for 1-2 days. It should start to show signs of bubbling and souring: fermentation.
Mix daily for up to 1 week, after which you can taste it. If it isn't soured enough or you wish to sour more, leave the ferment going for another few days or so, keeping an eye on it all the while. Once the ferment tastes good enough to you, strain through a fine sieve into a clean bowl, pressing on the fruit and peel pulp to extract as much liquid as possible. Keep in mind that the mangosteen ferment will have a red wine-like mouthfeel that can be a little strong, but mellows as it ages (or can be mellowed with soda/water dilutions later).
Pour into a bottle or jar, seal tightly, and leave out on the counter to become more acidic and fizzy, after which it should be consumed right away or the bottle burped and refrigerated.
Alternatively, simply bottle and refrigerate after decanting. The syrup can be kept in the fridge for up to two months.