While the tree is providing shade, the berries are messy. The flies are awful and I am constantly cleaning up the mulberries.
How to Keep a Mulberry Tree From Producing Fruit
Certified GKH Gardening Expert
I think what you have is a mulberry tree. It can be tricky to keep them from fruiting. There is no DIY method, but you can have a tree specialist come out and spray the tree with a sterilizing spray. It has to be done yearly, and typically the tree will still produce some fruit. It is just the amount will be greatly reduced.