This comes from a discussion in one of my history classes yesterday. According to the Anglo-Japanese Alliance, each nation had to assist the other if two or more nations declared war on them. The alliance was set up this way so that the UK and Japan wouldn't get dragged into each other's foreign policy constantly, but only when there was a large war with a coalition of nations arrayed against them. In 1905, Korea was pretty much a client state of Russia. When the Russo-Japanese War began, Korea initially started preparing its military to provide what little help they were capable of offering to their ally (Korea's military ranged somewhere between small and nonexistent at this point). However, Russia advised Korea to not join the war for two reasons (a) they didn't think they'd have too much trouble beating Japan and (b) they didn't want to risk activating the Anglo-Japanese Alliance and ending up at war with the UK. So what if Korea goes to war against Japan?
There are several ways this could occur. The Korean king had already proven that he wasn't the brightest guy in the world (If you doubt this, read about how the Sino-Japanese War got started. Pure stupidity on the part of the Korean government). So maybe he ignores the Russian advise out of some misplaced sense of patriotism. Or perhaps more plausibly, Russia doesn't advise him to stay out of the war, assuming that Korea can properly read the geopolitical situation and know that getting in is a bad idea for all involved (Korea gets curbstomped by Japan, and Russia is too busy with the UK to try to save them). Then the king proves that he isn't as smart as Russian leadership thought he was, and jumps into the war anyway.
So once Russia and Korea are at war with Japan, this activates the Anglo-Japanese Alliance, and the UK is bound by the terms of that alliance to go to war with Russia. What happens next? Would other nations (nations in Europe, USA, what's left of China, etc.) take sides in this war or stay out of it? Could this erupt into a WWI analog in 1905 (obviously with very different sides, but nevertheless)? Even if the war does remain limited in scope (UK and Japan vs. Russia and Korea), this could still have long term impacts on Central Asia, as the "Great Game" suddenly becomes a "Great War."