In a large bowl, combine mascarpone cheese and honey. Stir to combine. Add cream, milk, salt, vanilla and lime juice and whisk gently until completely blended together and no mascarpone bits remain. Cover and place in refrigerator to chill for 4 hours or overnight.
Meanwhile, make the blackberry puree. Add blackberries, corn syrup and sugar to a medium pot and stir to blend. If needed, use a potato masher or the back of a spoon to extract as much juice as possible. Simmer over low heat until the mixture has thickened and has a loose jam-like texture, approximately 15-20 minutes. Strain through a fine wire mesh sieve into a small bowl. Use a spatula or spoon to press as much of the blackberry sauce through the sieve as possible. Set the blackberry puree aside to cool completely. Once cool, cover and refrigerate until use.
Pour the chilled ice cream mixture into an ice cream maker and churn according to the manufacturer's instructions.
Transfer the churned ice cream to a large bowl. Working quickly, drizzle the blackberry puree over the ice cream and fold in using a spatula. Do not overmix - you want to have some visible variation in color. Transfer the ice cream to a chilled freezer safe container (I used a loaf pan) and cover with plastic wrap. Freeze for at least 4 hours before serving.