White Mulberry tree help...?

We have a very large White Mulberry tree in our yard (probably 50 ft. tall) that was here when we bought the house several years ago. While we are lucky that it only produces white fruit rather than the dark purple variety, it still creates a huge mess in our yard. We have contacted a few people about spraying the tree to reduce or eliminate the fruit production but no one will do it. I guess it must be illegal in Illinois or something since we can not even pay someone to have this done. Has anyone ever tried to catch the fruit in nets or something to eliminate some of the mess? Our backyard has almost become unusable in the summer which is a bigger deal now that we have kids. Between the mess of the slippery berries, there are also the hundreds of flies that are attracted to this mess as well as the raccoons that come to feast at night. I really do not want to cut the tree down for many reasons. a). We like trees b). It is a very large tree that keeps our house cooler in the summer because of the shade, c). It would cost thousands of dollars to have it removed. We have already paid about $700 a few years ago just to have a few of the large branches cut down to clean it up a bit. Last summer we noticed that the tree must have compensated from loosing the branches by producing even more fruit. We sweep our patio every day and have tried raking them from the lawn, but it is never enough. Anyone have any ideas?