Schubert calculus involves studying intersection problems among linear subspaces of C^n. A classical example of a Schubert problem is to find all 2-dimensional subspaces of C^4 which intersect 4 given 2-dimensional subspaces nontrivially (it turns out there are 2 of them). In the 1990's, B. and M. Shapiro conjectured that a certain family of Schubert problems has the remarkable property that all of its complex solutions are real. This conjecture inspired a lot of work in the area, including its proof by Mukhin-Tarasov-Varchenko in 2009. I will present a strengthening of this result which resolves some conjectures of Sottile, Eremenko, Mukhin-Tarasov, and myself, based on surprising connections with total positivity, the representation theory of symmetric groups, symmetric functions, and the KP hierarchy. This is joint work with Kevin Purbhoo.