Perhaps the first practical step in getting the two sides talking is in asking people to study the Frege-Husserl debate vis-a-vis psychologism creeping into thinking on the foundations of logic. Here were two great thinkers, both hard-science-oriented, both of the same generation and speaking the same language, both seeking the pathway to the grounds of objective knowledge, both anti-psychologistic, and yet not agreeing on what it means to be psychologistic at all. Remarkably, one is considered the beacon of the analytic school, while the other outlined the phenomenological project for philosophers and thus spurred the Continental school of thought. That is where the institutional schism began, since Frege influenced Russel deeply and Russel had enormous clout in the English speaking philosophical world, while Husserl taught Heidegger... and the rest is history. The study of the causal chain of affairs in Western academia in view of the rapidly widening idealogical gap is doable is reasonable time-frames.  

(By the way, Frege and Husserl were at least listening to each other. Two generations later Carnap was ridiculing Heidegger and the latter was ignoring him as superficial. In between, the Logical Positivists essentially castrated the notion of "Meaning" - and that I think is why speaking terms could not be established any more, since in philosophy, "Meaning" is way high up there with "Truth" and "Knowledge" in the pantheon of "Really Important Things"...)