I've seen RWD and AWD Civics out there. AWD Civics came from the factory, RWD did not.
If you really want this it'd be easier to just purchase an AWD Civic to begin with. The fourth gen actually saw a US Civic wagon with AWD. Not exactly anything to brag about though. Turtles are cool little cars anyway.
I've heard rumors that he drivelines aren't that strong, but there doesn't seem to be enough people out there to really say one way or another. I've never heard of specific problems with them.
I know a magazine or such put together an AWD Integra a few years back. I know it used CRV components and I believe it broke often.
A converted AWD or RWD Civic would be horrible for daily use at the very least. It would be something you would want to mostly leave at home and don't think it will win you anything besides some oohs and ahhs.