Existence of hyperbolic surface bundles with trace fields having real places

Let $S=S(g,p)$ be an orientable, connected surface of genus $g$ with $p$ punctures, and let $M$ be a hyperbolic surface bundle over $S^1$ with fiber of type $(-\chi,p)$, where $-\chi>1$. A trace field is the number field associated with the holonomy representation of $M$. Existence conjecture. For each pair $(-\chi,p)$ with $-\chi>1$, there exists a hyperbolic mapping torus of type $(-\chi,p)$ with trace field having a real place. This conjecture extends the study of real places in trace fields beyond once-punctured torus bundles, for which Calegari proved that no real places occur. The paper exhibits several infinite families of such pairs, but does not establish the assertion for every pair.