[dice]

I've seen pictures of Japanese Tomato Rings. The plants
are generally planted around the outside, with compost
on the inside. I seem to recall that growers tie the plants
to the fence as they grow, so that the stems will be in close
contact with the compost inside the ring (that way they can
sense that there is something like soil there to root into).

Even if you made the ring as tall as a typical CRW cage,
with compost in the bottom 3' of it or so, the tomatoes
still would not thread themselves through it. They would
probably flop together in the middle, and sprawl on top
of the compost pile in it.

About the only maintenance-free method of trellising
is the cylindrical cage that surrounds the whole plant
and is as tall as it needs to be. (A-frames are close, but
they still need a little work because of the open sides.)
Tomatoes are only "sort of" vines. They are not climbers,
so they do not naturally weave themselves through a trellis
as they grow, you have to do it for them.

Anything else I might suggest would be entirely experimental
(like sinking 4 posts per row, 2 at each corner, and securing
successive horizontal layers of galvanized hog fence or
nylon fish net to them at 2' intervals as you go up the posts;
you probably still want to anchor the posts past the ends
of the row to keep them from sagging inward toward the
center of the row as weight builds up on the horizontal
layers of support).

Edit:

(They would flop together in the middle if you planted them
*inside* the tomato ring, which is not how they are normally
used.)

"... 4 posts per row, 2 at each corner..." should be
"... 4 posts per row, 1 at each corner ....".