Infinite monochromatic sumsets for colourings of the reals

N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $\mathbb{R}$ so that no infinite sumset $X+X=\{x+y:x,y\in X\}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:\mathbb{R}\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb{R}$ so that $c$ is constant on $X+X$.