In a small saucepan on the stove top, combine water, sugar, and corn syrup. Stir the sides of the pan until the sugar dissolves, brushing down crystals with a damp brush if needed. Heat over medium-high heat until the mixture reaches 240°F (soft-ball stage) on a candy thermometer or instant-read thermometer.
Meanwhile, in the bowl of a stand mixer fitted with a whisk attachment, beat the egg whites and cream of tartar until soft peaks form. Scrape down the sides of the bowl as needed.
With mixer running on low mixer speed, slowly stream the hot syrup down the side of the bowl (avoid the whisk). Once all the syrup is added, increase to medium-high speed and beat until thick, glossy, and fluffy—about 7–8 minutes—until the mixture holds glossy peaks or stiff peaks.
Beat in the vanilla extract just until combined.
Transfer the finished homemade marshmallow cream to an airtight container.