Answer: Ficus (fig) trees are grown for edible fruit or rubber production in many tropical climates. Based upon your description of fruit location on the trunk, it sounds as though your ornamental fig is Ficus racemosa. These plants are commonly grown in the mild-winter areas of southern California. Figs are unusual fruits in that they form and grow from flowers that are opposite in appearance to most fruit-bearing trees. They're inside-out! Because of this it's difficult to keep fruit from forming. Sometimes a growth inhibiting hormone can be sprayed on the open blossoms of fruit trees to keep fruit from forming. But, with your fig tree, there's no practical way to apply growth inhibiting hormones.
It may be than you'll either have to put up with the messy fruits, or you'll have to remove the fruit as it forms, but before it matures enough to ooze sticky sap.
Wish there were more to suggest!
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