Actually, the covenants do allow for small, short fences in the back that are attached to the house. I have seen several in my travels around the Villages (e.g. Sunset Point Blvd just before you get to Heron and Pelican golf courses on the right hand side). The ones I have seen are no more than 3-4 ft high, white picket, appear to be vinyl. They can't extend beyond the side perimeter of the house and must adjoin the house (i.e. can't be seen from the front). I'm certain you would need to submit your design to the architectural review folks for approval.
So, it can be done.... I don't know all the limitations on size, materials, etc.
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Maryland (DC Suburbs) - first 51 years 
The Villages - next 51 years 
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