Typically the right to equal possession is a temporary arrangement while the parties work to resolve any matters which have arisen as a result of the breakdown of the relationship. Usually the only way that equal possession ends is through settlement of the matter and the establishment of a separation agreement. If your matter goes through the court system then one party can be granted exclusive possession on a temporary basis by way of a court order.
You can usually only get a court order for exclusive possession of the home if there is a good reason why one of the parties should not be allowed to remain in the home. These situations include situations in which there are crimes being committed or activities which are dangerous to others in the home taking place in the home caused solely by one spouse. Additionally this includes cases in which your spouse is purposely damaging the home.