Stir mayonnaise, tomato paste, red onion, green bell pepper, red bell pepper, lemon juice, and salt together in a bowl. Cover the Andalouse sauce with plastic wrap and refrigerate 8 hours to overnight.

Heat oil in a deep-fryer or large saucepan to 300 degrees F (150 degrees C).

Submerge potato strips in a bowl of ice water; rinse potatoes in several changes of cold water until water in the bowl is clear. Drain potatoes and pat dry.

Working in batches, fry potatoes in the hot oil until just cooked through but still white, about 5 minutes. Transfer fries to a paper-towel lined plate to drain. Bring oil back to temperature between batches. Cool potatoes to room temperature, about 30 minutes.