I don't like REITs because they are more closely correlated to the movement of the stock market. They don't really do the job of diversifying a portfolio because of that correlation.
When the stock market dropped in 2008, REITs were hammered as well because the housing bubble burst. Bonds went up, and if you rebalanced (sold the bonds to buy more stock) then you came out much further ahead when the stock market recovered.
The point of adding bonds for diversification is that they move in the opposite direction of equities; blunting the major drops (and providing buying opportunities). REITs don't fit that bill.
REITs are not undergoing a correction like bonds because the price of real estate is a function of housing supply and buyer demand. Rising interest rates only make it a little harder for buyers to buy, so the effect of rising interest rates on real estate prices is muted. The other effects on real estate prices (more wealth in the economy for buyers) pushes in the opposite direction of the rising interest rates.