Dry Rot/Wet Rot?

[The Diver]

I have come across some dry rot in jarrah pergola timbers and am going to remove and replace. The rot is only in the battens and not in the main beams as yet. I had painted the timbers with solar guard green paint when built. The structure is covered in 80% weave shade cloth. My question is .... does anyone know what to treat the existing beams with, to prevent further spread of the rot? I read in one of the forums about applying bleach to kill the fungus - is this effective? Is there a special paint to use to inhibit spread?
Thanks
Greg