While not exactly the same, I have see telephone poles in extremely wet areas buried into the ground as usual but with another pole bolted to it horizontally along ground level. Guy wire(s) to a solid location on the 90 degree axis do the rest. In an area where it needs to remain a bit more picturesque, they will bury the attached horizontal pole just under the surface so it is not seen.
I suspect you don't need, nor would you want, guy wires, but burying some horizontal timber attached to the fence posts might help it from sinking or tipping. You could even use two in an X pattern. That might be overkill though as the structural integrity of the fence itself would prevent any lean along the axis parallel to the fence.
