Ooh... I would be so tempted so that the chicken run went up against the garden fence.
In my setup the coop is in the middle of a run complex that shares a fence with my orchard and another with my vegetable garden. Now, my orchard didn't really work out (sniff and sigh) my elevation is just a bit too high, and I have a couple of apple trees that are still alive, but I have had them now long enough that it is very clear that they will never grow much and will never bear fruit (well, maybe one apple five years from now). The chickens would eat apple leaves, so I just put tiny fences around the trees, and that worked. My orchard and vegetable garden both have dog and moose proof fencing. I put netting over the top of both to keep out raptors. I LOVE the fact that I can let my chickens into my garden. I used to have a crop of weeds up before it was warm enough to put out seedlings, and it was a great deal of work to clean up the beds. No longer! The chickens clean it up beautifully for me, and even make the beds nice and fluffy from their scratching.
Things to watch out for.....
1. The much cheaper and easier to instal fishnet does have some drawbacks. I have the plastic one over my garden, and it is light enough that I was able to staple it to the wood fence that surrounds the garden and not have any support posts in the middle. I think my garden is, at a guess, 24 or 30 feet square. I did have a raptor dive right through the same net over my chicken runs, and kill some chickens, but I think because the garden has more cover, I haven't had that problem there (yet).
2. Those chicken necks stretch lots farther than you think they would! If the chickens can get their head through the fence they can reach pretty far in to eat the produce, and I tend to use my garden fence to support peas etc. I have plastic corrugated paneling up against the fence that supports my peas, it helps to reduce wind and make things warmer (which I need) and also keeps those chicken heads at bay.
3. Even though this has worked wonderfully well for me, I only have raptors (tons) and dogs to worry about. Well, supposedly weasels and bears, but neither of them have found my coop.
But I also love having the chickens right on the other side of the fence, I toss them slugs when I find them, and lots of the weeds, so even if you don't want to go through the extra trouble of setting up the garden so they can use it as a run during garden clean up time, it is nice to share a fence with them to make tossing them treats easier. It makes weeding a super happy time for me. Anything that makes weeding fun is SO VERY GOOD!