An algebraic variety is said to be rational if it is birational to projective space. In this talk, we study the rationality question over the real numbers for a certain class of conic bundle threefolds. The varieties we consider all become rational over the complex numbers, but in general the complex rationality construction need not descend to R. We discuss rationality obstructions coming from intermediate Jacobian torsors and from the real loci of these varieties. This talk is based on joint work with S. Frei, S. Sankar, B. Viray, and I. Vogt, and on joint work with M. Ji.