How many palms are there? As far as the fronds are concerned, you are allowed to remove anything that crosses your property line, just be sure you know EXACTLY where that line is. Caution, though. If you remove all the roots on your side, you may destabilize the palm and make it move likely to fall on your side of the property during a hurricane. You can also report him to the city Property Standards office if he refuses to remove dead fronds.

Is there some way you can build a protective covering over your garden that will keep the fronds from falling on it? Maybe a lattice arbor to catch the fronds? I know sometimes city ordinances can be crazy about building things like that, but if you say you want to do it to keep the fronds from damaging your property, they may show lenience..

Of course, you can always just kill the palms, not that I'm recommending that, but Royal Palms are susceptible to a lot of diseases.

Legally you can not kill your neighbors trees nor cut the roots that will cause the trees to die. Palm roots don't usually crack concrete so i would say a burried wall of concrete would stop the invasion but as dirtygardener pointed out if you trim back the roots too severely you can cause the tree to fall your way during the wind. So i would say be cautious.

Yeah I agree with dirtygardener and Pea about not removing the palms roots. Royal palms will never be up there with Australian pine and Brazilian pepper because unlike those two, Roystonea regia ( Royal Palm ) is native to south Florida. It also doesn't self seed aggressively or sucker up like the former two.