, nice to meet you and howdy neighbor!
Ordinarily, I'm with @Mystriss
, and would do what I can to save a tree.
But that tree looks pretty bad. In particular that big long crack along its length. Once that happens, it will likely not every fully regain its structural integrity, which means it might come crashing down in a windstorm, though in much of California, that usually happens in the winter when mulberries are bare.
So, I concur with @Bondo
and @Old Thomas
and advise immediate chain saw therapy, with possible replacement.
While Mulberries are fast growing, they're not the total trash that some other fast growing trees are. They offer very fast growth, a wide crown and lots of shade in the summer, but sun in the winter, which might be what you want. As a practical matter, that means that planting a new mulberry and taking reasonable care of it is a reasonable option. You're talking a relatively short time till shade.
Just out of curiosity, where in California are you? I'm in OC/Whittier area. I ask because, depending on where you're at, I might be able to recommend a replacement if you want one.
And, mulberry appears to make reasonably good barbeque wood. No sub for mesquite to be sure, but it's a hardwood, and once dry, lights easily.
Hope this helps.