Place the egg whites into the bowl of a stand mixer along with the cream of tartar, and mix on medium until firm peaks form.
In a small saucepan, combine water, sugar, and simple syrup or light corn syrup. Place over medium-high heat, clip a candy thermometer (best to use a digital thermometer if you have one) onto the side of the pan, and continue to cook until the mixture reaches 240˚F, approximately 7 to 8 minutes. Once the mixture reaches this temperature, immediately remove from heat.
Turn the mixer on low speed and, while running, slowly pour the sugar syrup down the side of the bowl into the egg whites. Make sure this is poured slowly; if poured too fast, you could cook the egg whites. Once you have added all the syrup, increase the speed to medium.
Continue to whip until the mixture becomes very thick and glossy, approximately 10 to 12 minutes. Add the vanilla bean paste during the last minute of whipping.
When ready, pour the marshmallow mixture into a container to use at your leisure. You can store this in your refrigerator for up to 2 weeks, if it lasts that long.