Peel and finely grate the garlic cloves. Remove the rosemary leaves from the sprig and chop them finely.
In a small saucepan, heat the olive oil over medium heat. Add the grated garlic and chopped rosemary, and sauté for 1-2 minutes, until fragrant.
Pour in the fresh lime juice, honey, and rice vinegar. Stir well to combine, and let it simmer over low heat for 3-4 minutes, until slightly thickened.
Season with salt and pepper, then remove from heat. For a smoother consistency, blend with an immersion blender.
Let cool slightly, then serve with grilled meats, fish, or salads. It can be stored in the refrigerator for up to a week.
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