It's a bit early but I went on a micro fig hunt with Art (kubota1). I say micro because it was only one fig tree. The tree was referred to me from a friend of a friend of a friend kind of situation. This person has it growing in his house and really doesn't know a single thing about it. He's not into figs, it's just a plant to him.
To start he:
doesn't remember where it came from,
doesn't know the variety,
doesn't know the color of fruit etc. etc.
The tree is in a 10 gallon pot and is growing horizontal like an octopus. I think the reason it's growing flat is because it's starving for light and trying to reach sunlight from the nearby window. He, on the other hand says it grows like that because it's a dwarf tree. I tried to tell him that it needs light but in the end I just agreed with him since there's no chance of changing his mind.
Art and I did manage to get a couple of small cuttings from him. I know it's a long shot but would anyone be able to start the pinpointing of what variety it is? All I can show you is the leaf pattern, which it is a large leaf, bigger than your hand.
Any guesses would be helpful.

Never seen anything like it. Octopus describes it exactly. About 2 ft. tall and 10 wide.
Dang thing was taking up so much space.

Reminds of Magnolia

IDing isn't my strong suit but I have a White Paradiso that constantly wants to grow horizontally and has leaves close to that if my memory holds. Joe at fig trees.net sells these and may have pictures. I think he has them labeled Paradiso or Genova. 
If it is Paradiso it's a nice fig that looks a lot like Brunswick. If I put mine in a hanging pot and let it it would become a weeping fig. After 6 years of pruning it still insists it's not a tree but rather a hanging fuchsia that just happens to produce figs. They are very tasty figs but will crack if over watered. I'm not sure how many types of Paradiso exist. I have 2 as I also have Paradiso "Gene". I remember reading somewhere (sorry) that there are at least 6 different Paradisos, but . . . 
Good luck.

I'll have to try to get back there to see the figs when they ripen to get a better ID on it. I never seen a tree growing like that. I don't know where I would store something like that for winter as it takes up a lot of space. His tree is spread out a good 5 feet from the center on one side and 2 to 3 feet on the other side. He's about at his limit that's why we got a couple of cuttings. It will be interesting to see how mine grows. If it spreads then I'll try to train it to grow upright. Might be good for someone who has a warmer climate and the room.

Your description sort of fits a description I copied from somewhere either here or gardenweb for one of the figs I am currently rooting.  I wish I could give credit to the person that posted the following: 

"Pierre Baud describes Madeleine de deaux saisons to Brunswick. I can certainly see that.  This is an interesting little tree.  The tree spreads out like a spider plant.  It's in its second crop now.  The fruit on this tree can fool you!  The bottom half is red but the top half is light green.  You have to pick it when it is half and half.  I learned the hard way after one bird pecked the bottom just before I picked it!  It has amazing red pulp and strong sweet flavor.  I got mine growing in superroot pots and it is doing fine.  This tree also loves the heat!"

I wish I knew which post I took that from, but someone said the above.  I always keep descriptions I find for all the figs I am growing or rooting to give me an idea of how they will perform.  I didn't keep the author's name for the above :-(

Suzi

Suzi, that's an interesting description. Makes more interested in this tree now. I'll have to post the results when I find out.
Things that make you go hmmmmmmm!