The death of Yi Sun-sin sometime prior to Toyotomi Hideyoshi's invasion of Korea leads to Japanese victory. This, in turn, promotes an expansionist, rather than isolationist, Japan. Under the leadership of the Toyotomi family, holding the court titles of Quampaku and Sexxô, the Japanese compete with the great European powers for colonies.
Japan and her former colonies form a large, if loose, union covering large areas of the world. The growth of the Empire began as they expanded Northward to Siberia and then westward until colliding with the Russians, and across the Bering Straits into Aresuka, and down the Pacific coast. They also expanded into Southeast Asia, Indonesia, and into Australia (Xinnoranda).
Today, most Japanese colonies retain at least a nominal connection to Japan proper.
In the Americas, Christian and Buddhist missionaries came into frequent contact, sometimes resulting in violence, but eventually the two religions came to co-exist relatively peacefully. Japan's involvement in the world meant that Christianity was moderately successful in Japan, though still a minority religion, while Buddhist missionaries made their way into Europe.