Mix crumbs and butter together and lightly press into 9 inch square baking dish. Refrigerate while making filling.
Beat cream cheese, marshmallow crème, lime juice, and vanilla with electric mixer until light and fluffy. Gently fold in fresh raspberries and spoon filling over crust.
Combine the sugar and water in a medium heavy pot and bring to a boil over high heat. Reduce the heat to medium-low and simmer until the sugar dissolves, about 2 minutes. Add the raspberries and cook, stirring occasionally, until the syrup thickens and reduces by 3/4 in volume, about 5 minutes.
Use a flexible rubber spatula to push the raspberries through a fine mesh strainer. Press down to extract as much juice as possible. Continue to press until all that remains is seeds.
Stir in lime juice to taste and cool.
Spoon the coulis in a plastic bag. Snip small corner of plastic bag and drizzle the coulis over the raspberry cheesecake filling in a zig zag pattern. For a decorative effect, run a knife through coulis, in opposite directions.
Any leftover coulis can be refrigerated in an airtight container for up to 3 days.
Cover and refrigerate at least 4 hours, but preferably overnight.