The answer, like most things, is 'it depends'. In this case, it depends why rates are rising. If the economy is doing well, economic activity will force inflation to rise, and interest rates will rise accordingly. In this case, like mathjak says, prices will be more a matter of supply vs. demand. A strong economy drives job creation and wages rise, this is a positive for real estate. Now, on the other hand, if rates rise for other reasons...say because interest rates have been artificially depressed by central banks around the world and they decide to try and correct that...then the rise in interest rates is not driven by a rapidly growing economy. In this case, there are not lots of new jobs created and lots of pay increases. In this case, the rate rise effects real estate prices just like they affect a bond...rates up, prices down.
In reality, it will be a mix of both and your local community will be impacted by local matters as well.
FWIW...the general rule is REITs generate most of their price returns during stable/declining interest rate environments, while remaining stable or declining during rising interest rate environments.