Put the juice in a small non-reactive saucepan and bring to a boil. Cook until reduced to 1/2-cup. Let cool to room temperature. To accelerate the cooling process, pour juice into a small bowl and place the bowl into a larger bowl with 2 cups of ice and 1 cup of water in the bottom.
Combine half of the reduced juice, tarragon and egg yolk in a blender and blend until well mixed. With the machine running, slowly add the olive oil in a thin, steady stream. Blend until the mixture emulsifies and becomes thick. Taste for seasoning. Add more tangerine juice if a thinner mayonnaise is desired. Scrape into a jar, cover and refrigerate until needed (keeps 2-3 days if refrigerated).