You will need to authorize your out-of-state LLC to do business in Florida if you are transacting business in Florida. If the entity is collecting rental income, then the LLC is considered to be transacting business in Florida and will need to be authorized to do business in Florida.
It should be noted that owning real property, in and of itself, is not considered transacting business in Florida. See Florida Statute 605.0905(1)(m) (2024). Thus, if you own real property in Florida with an out-of-state LLC, but the property is not collecting rental income or otherwise transacting business in Florida, then the out-of-state LLC is not required to be authorized to do business in Florida.