It also depends on what you mean by "break them up", and they do need to be cut pretty carefully so that you have the potential for eyes on each tuber you save.
Mine have a label tied around the stalk of each plant and buried with them when I plant them out, so there isn't any mix up when I dig them in the fall (theoretically) Generally they are wet enough when I dig them that sitting on the concrete floor of my (barely heated) work room for a couple of weeks as I get around to dividing them isn't a problem.
As I divide each clump I write the name with a sharpie on each tuber then put the groups in little lunch sized paper bags also labeled. Then when I get them all done, I dust each group with cinnamon for help with fungus/,mold prevention, and then wrap the individual tubers in saran wrap, pop them back in the paper bags, and store the bags in boxes on the floor of my cold work room for the winter.
It takes way to long, but some how its worth it.