Legalized polygamy could create a lot interesting legal issues, particularly marriages of convenience. It would challenge a lot of traditional legal structures that assume a union between two people. If there is no restriction on how many people you can marry at once then you can start giving out residence rights via marriage or multiple wealthy individuals can amass large estates via marriage. I am sure there is a lot more.

This is not meant as an argument against legalized polygamy, rather to point out some of the obstacles it faces.