Make the babka dough. Heat the milk in the microwave until it reaches 100 degrees F, about 30 seconds. Add the yeast and 1 teaspoon sugar and let sit for 5 minutes or until the mixture is foamy.
Combine the yeast mixture with all-purpose flour, sugar, salt, vanilla, and egg. Mix until the dough comes and knead for 2-3 minutes or until firm. Add the butter a little at a time, mixing well between each addition. Continue to knead the dough until smooth, about 5-6 minutes.
Transfer the dough to a clean bowl, cover, and let proof in a warm area for 1 hour. It may not double in size.
Meanwhile, make the raspberry lychee rose jam. Puree the raspberries and lychee in a blender or food processor until smooth. Combine the raspberries, lychee, sugar, and lime juice in a medium saucepot. Bring to a boil over medium-high heat. Reduce heat to medium and continue to simmer for 20 minutes or until slightly thickened. Remove from heat and stir in the rose syrup. Let cool completely.
Grease a 9x5 loaf pan and set it aside.
Punch down the dough. Lightly dust a clean work counter with flour and roll out the dough until it's about 9x18 inches. Spread the raspberry lychee rose compote in an even layer. With the long side facing you, roll up the dough into a log. Place the seam side down and use a serrated knife to cut the log in half.
Flip the logs so that the cut sides are facing up. Gently pinch the ends of the top of the log together and start to twist the two logs together. Lift one side over the next to form a twist, keeping the cut sides facing up. Pinch together the ends and transfer the log to the prepared pan. Cover the pan, and let rise in a warm place for 45 minutes.
Preheat oven to 350 degrees F.
Bake the raspberry lychee rose babka for 1 hour or until golden brown. If the bread starts to brown too quickly, loosely cover the top with aluminum foil. Remove from the oven and cool. Serve warm or at room temperature.