Make the guava coulis: combine all ingredients in a non-reactive saucepan and cook until the sugar is completely dissolved and the mixture comes to a simmer. Remove from the heat and transfer to a non-reactive container. Chill thoroughly.
Preheat the oven to 325 degrees F.
Make the crust by combining the crumbs, sugar, and melted butter in a medium bowl and stir to mix well. Press the crumb mixture onto the bottom and slightly up the sides of a 9-inch springform pan. Bake until golden brown and fragrant, about 10 minutes. Transfer to a wire rack to cool while you prepare the cheesecake filling.
Combine the pineapple, pineapple juice, and lime juice in a medium saucepan and cook over low heat until most of the liquid has reduced and the pineapple is very tender, about 20 minutes. Be careful not to let the pineapple burn towards the end of the cooking time. Remove from the heat and puree until very smooth. Chill thoroughly before proceeding. You should have about 1 cup of puree.
In the bowl of an electric mixer combine the cream cheese and sugar until very smooth, stopping to scrape down the sides of the bowl as necessary. Add the eggs 1 at a time, mixing thoroughly between additions. Add the pineapple puree and vanilla. Transfer the mixture to the prepared springform pan and bake until the cheesecake is set in the center and lightly golden around the edges, 55 minutes to 1 hour. Carefully remove from the oven and transfer to a wire rack to cool. Refrigerate at least 4 hours and up to overnight before serving. Garnish each slice of cheesecake with some of the guava coulis and chopped crystallized ginger.
Note: The guava coulis and pineapple puree used in this recipe may be made up to 1 week in advance and stored in non-reactive, airtight containers in the refrigerator.