Dice the ham into small 1/2" cubes. Add all of the remaining ingredients except for the relish to a food processor and pulse until chunky. (You want it to be smooth enough to spread, but not so smooth that it becomes a paste. Check every few pulses and watch closely to get the desired consistency).
Transfer the spread to a mixing bowl and stir in relish. Serve on white or rye bread or with crackers and Swiss cheese.