On ncatlab page on formality, it is stated that Deligne--Griffiths--Morgan--Sullivan proved that the real homotopy type of a closed Kaehler manifold is formal. Later, Sullivan "improved" this to $\mathbb{Q}$-formality. 


My question is: are there some easy examples of closed topological manifolds whose $\mathbb{R}$-homotopy type is formal, but $\mathbb{Q}$-homotopy type isn't?



Formality over $\mathbb{R}$ vs formality over $\mathbb{Q}$
What Sullivan proved is not just that the $\mathbb R$-formality from Deligne-Griffiths-Morgan-Sullivan can be improved to $\mathbb Q$-formality, but rather that formality over any field of characteristic zero for any space always implies formality over $\mathbb Q$. See Sullivan's paper.


