An algebraic variety over a field k is said to be rational if its function field is a purely transcendental extension of k. In this talk, we work over the real numbers and study the rationality question for a class of conic bundle threefolds; the varieties we consider all become rational over the complex numbers, but this rationality construction does not in general descend to R. This talk is based on joint work with Sarah Frei, Soumya Sankar, Bianca Viray, and Isabel Vogt, and on joint work with Mattie Ji.