Are there speakers with cloth/soft enclosures instead of wood?

What does a soft enclosure sound like?
Sounds like mush. At least the bed board ones I made did.
Good enough to cover a squeaky bed.

Um, well the enclosure serves two purposes, keeping the front and back waves apart as well as providing part of the suspension.
There's open baffle, and some speakers rely heavily on thick foam as kind of wave guides.  
Assuming you made a baffle of say, organic Egyptian cotton, it would just blow back and forth, and essentially the speaker would be in free air.
SpeakerLab tried a cloth speaker back in the 90's. Put a lot of money into it. Got hung out to dry. 
There was the Acoustic Precission Eikos FR-1 produced some time in the 90s and the result of a collaboration between Tom Evans, Patrick Handscombe (presumably not the cricketer) and Ted Jordan in the UK. It was a single driver in a polystyrene foam box with a thin skin on the outside. It worked very well when I heard it at Tom’s place in Wales and my local hifi shop carried it for a while but I don’t think it sold that well. People prefer something more substantial for their money it seems. The finish wasn’t very robust I’ve heard. 
So - speaker in free air = one hand clapping? On the other hand (!) a box is a problem too? Sorry. Home too long.
On the same note, why wood? Why not granite, cement or something super-stable? I remember when that was the fad.
Vandersteen Model 1,2, and 3

Yep, and my 2CE Sigs sound fantastic. The cloth covering is also part of the acoustic design and engineering.

Yes, the actual speaker drivers are in a box behind the cloth, but that is rather limited for the mid-range and tweeter, not so much the bass driver and acoustic coupler; Providing for sound coming out around the speaker, unlike a conventional box. Somewhat between an open baffle speaker and a true box speaker.
“ none can free us, save our own minds’ “

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please someone report this as a mental agility test for overzealous moderators...
I've never heard Vandersteen speakers, but I will make sure I do before my next speaker purchase. Very intriguing. 
Vandersteen Model 2 has sold over 100K pairs. One of best selling speakers of all time.
The fabric of our lives. 
The latest textile tech.
Don't pull the wool over our ears.
Magico carpet.
Who Polked a hole in my shirt?

Yes there are and they are the best, IE, the most accurate speakers made.  Look up Magnepan and give your favorite model a tryout in your room.

Any fully open back speaker could qualify.  There are a lot:  Electrostatics, Maggies, Time Windows... yes they have wood frames, other wise they would droop and sag.  
Don't forget Vandersteen 4's. All you see is the top & bottom oak caps & that big black sock wrapping 95% of the speaker. 

I've seen pictures of the Vandy 4s sans sock. Very odd looking thing: a very strong central tower of various speaker enclosures (black MDF) attached at top & bottom.
Magnepan, Vanderstein, Dalquist, Thiel, Heil, Ohm all made(make) speakers at least partially covered by fabric (although all but Magnepan,  Quad, and Dalquist had boxes for some of the drivers.)  Going back even further, Stu Hegeman had a partial fabric covered speaker sold under the Eico and the Harmon-Kardon labels.
Back in 1980 I made speakers for my basement, one of many systems I have. The tweeters and mid-ranges drivers were placed in spherical polystyrene bead board enclosures. The eight-inch woofers were in hollow spherical enclosures made from particle board and covered with 1/4" felt. I suspended these six time-aligned speakers from the ceiling using ropes. For the sub woofer I built a "big foot" with a 15" woofer from an article in Audio. It is the size of a refrigerator and I laid it on it's back. It was electronically crossed over at 45 Hz. I still listen to this system and have a Yamaha PSR 190 keyboard hooked up to it. The six speakers use a custom-made passive crossover from recommended parts I got from Bob Ludwig of St. Louis. I currently drive the six speakers with a McIntosh 2300 solid state power amp and the sub with a Hsu class AB sub amp rated at 250 W per channel. The spherical polystyrene bead board enclosures used for the tweeters and mid-ranges are quite non-resonant. The system sounds well balanced.