Special equipment: a deep-fry thermometer
For the French fries: In a deep pot or a tabletop fryer, heat the vegetable oil to 250 degrees F.
Cut the potatoes into batons leaving the skin on (the pieces should be 1/4 inch by 1/4 inch by the full length of the potatoes). Rinse the potatoes under cold water.
Fry the potatoes, in batches if necessary, until just tender but not mushy, about 4 minutes. There should be little to no color. Transfer the blanched fries onto a baking sheet and cool completely in the refrigerator.
For the tartar sauce: Combine the mayonnaise, buttermilk, capers and relish in a bowl and whisk together.
For the fried fish: Increase the heat under the pot of vegetable oil and heat the oil to 350 degrees F.
In a bowl, combine the eggs, soda water and crushed ice. Add the flour and whisk the batter. Do not overmix. Place the panko in a second shallow bowl. Place some flour for dedging in a third shallow bowl.
Season the fish with salt and pepper. Dredge each piece in the flour, then dip in the batter and then dredge in the panko.
Deep fry the pieces of fish, in batches if necessary, until golden brown and crispy, 4 to 6 minutes. Remove from the fryer and allow to rest on a cooling rack for 5 to 10 minutes.
While the fish rests, finish cooking the French fries. Bring the oil temperature back to 350 degrees F. Fry the French fries, in batches if necessary, until golden brown and crispy, 4 to 5 minutes. Toss the French fries in a bowl and season with salt and pepper.
For plating: Mound some fries on each plate and place a piece of fish over the top. Serve the tartar sauce on side.