Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce the heat to medium-low and simmer until tender, about 20 minutes. Drain, then cool, peel, and chop.
While the potatoes are cooking, place eggs into a saucepan and cover with water. Bring to a boil, then remove from the heat and let stand for 15 minutes. Remove eggs from the hot water, cool under cold running water, and peel. Chop the whites; discard the yolks or save for another use.
Combine potatoes, egg whites, celery, onion, relish, mayonnaise, mustard, celery seed, garlic, and pepper in a large bowl. Mix together well and refrigerate until chilled.