Prepare the ingredients: juice the limes, zest the peel, finely chop the red onion and garlic, and chop the tarragon.
In a medium saucepan, heat the olive oil over medium heat. Add the chopped red onion and sauté for 3-4 minutes, until translucent.
Add the chopped garlic and sauté for another 1 minute, until fragrant.
Pour in the tequila and simmer for 2-3 minutes to allow the alcohol to evaporate.
Add the freshly squeezed lime juice, lime zest, apple cider vinegar, and brown sugar. Stir well to combine.
Season with salt and pepper, then bring the chutney to a simmer. Reduce the heat to low and cook for 15-20 minutes, until it thickens to a jam-like consistency.
Add the chopped tarragon and cook for another 2-3 minutes, allowing the flavors to meld.
Let the chutney cool completely, then transfer it to clean jars. Store in the refrigerator for up to a week.
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