Oh my gosh YES!!!!
I so want to come help! I really do.
This would be sooooo easy if you use a flexible PVC piping. Simply insert into the top of the fence and arc it over to the other side. Of course you would have to determine the arc and the length of PVC but that is all just "working it out". I say go buy two lengths of 10 or 12 feet with a female to male coupling between them; probably 3/4 inch to 1 inch depending on the inside diameter needed to insert over the fence spikes.
After several arc supports across whichever direction you prefer (from the looks of it your garden is wider on one side and would be a nice big arc from side to side [widest]). Then simply take a drill and drill through each arc at measured, marked spots and attach a cross beam made out of more PVC or a smaller stock of 1/2 x 1 wood (I think this is like .98 cents a length of 6ft.) This will make it all sturdy.
Additional supports would need to be worked in based on any 'play'.
Then use a heavy mil plastic. If you are going to use this over several seasons I would look into something thicker than what is sold in the home improvement stores. Several places online have thick heavy gauge plastic for good pricing.
The plastic can be placed right over your existing fence and PVC framework and then secured with button ties (homemade I suppose) with a piece of duct tape or aluminum tape for strength and resistance to tearing through. The seams between each sheet of plastic can be fastened with the aluminum tape as well.
A button tie can be made with a those tin/aluminum disks used with roofing nails to hold down sheets of roofing paper. Simply make two holes in the center and then thread these with either nylon/sinew/or wire, then tie on the inside of the structure. Then cover the button with a piece of aluminum tape to make it adhere and not allow the plastic to pull and rip away.
It is wise to have a cross flow of air on the hottest days (best to measure inside temps). One being the doorway and the other the backside of the enclosure like a prop-up window of sorts; something you can close and prevent cold air from entering. On those hot days simply walk out there open the door and also the prop-up window; place it high so this can allow for any overheated air to escape.
Of course be sure and put something to sit on in there with a little table, because there will be those times that you will find sitting amongst the humid air of the plants with a nice book in the winter is a wonderful feeling.
If you feel that you might have very low low temps you could put a 55 gallon drum and fill it with water. The water will take on the daytime temperatures and through the night it will expel this warmth (or warmer than the night air) and keep things from freezing.
Or during those late night freezes simply string out an extension cord and hang a ShopLight in there. You will be surprised how a 40-60 watt bulb will really heat up an enclosed space.
...OK, I got my bags packed, here I come
I have a feeling this is temporary until Spring and then you will recreate your open garden again. I would say that with the right ingenuity this can be made to come down with just as much ease.
I cannot wait to see pictures of what you decide to do.
Ron