Operation Pukchin Tongil
By 1955, the military balance between the two Koreas had grown even more lopsided. Kim il-Sung simply was engaged in a variety of factional struggles with his Soviet-backed military advisers - and simply didn't trust the Korean People's ARmy enough to funnel massive amounts of funding into it. In contrast, President Lee Beom-Seok had funneled untold funds into the new Korean National Army. Despite its military inferiority, the North Koreans attempted to play tough by regularly participating in brief incursions into the DKR to support left-wing guerillas and rebels. This had the negative effect of encouraging the anticommunist powers to funnel even more funding into the KNA.
President Lee could not help but notice that Soviet forces in the Far East had almost entirely been shuffled to the West to deal with various European crisises, such as Yugoslavia, Finland, Poland, and to a lesser extent, Hungary. In particular, the Soviet Fifth Army moved West on the request of Field Marshal Rokossovsky to suppress the Polish Revolution. Lee contacted all of the local regional anticommunist powers - including President Kai-Shek of China, President Richard Russell of the USA, and Prime Minister Hatoyama of Japan. Chiang and Russell enthusiastically endorsed his plan, Hatoyama actually disapproved, but he was the least important actor here. Chiang was still smarting over the Xinjiang coup and saw an easy way to strike back at the USSR - while Russell was elated to find a way to strike back at the USSR for their Yugoslav invasion that didn't involve directly going to war against the USSR.
On the morning of March 1st, 1956, the 37th anniversary of the famous March 1st Movement for Korean independence, the Korean National Army launched a wide sweeping attack across the border. On the eve of war, the North Korean Army numbered roughly 132,000 men, largely concentrated in defensive positions across the border. Although Kim il-Sung asked Joseph Stalin for permission to invade the South in 1950, Stalin turned down his request, not excited about getting dragged into an East Asian War that he just had narrowly avoided in China. However, he never gave up on the idea, and placed KPA divisions on the border ready for a strike south. In contrast, the KNA numbered roughly 250,000 men. Both sides were well-armed due to receiving surplus weapons from the USSR and USA, which ramped up in the early 1950's as both armies demobilized from World War II. In fact, on the eve of the war, both armies greatly resembled in equipment their patrons - the North Koreans sporting Mosin-Nagant rifles, T-34s, and Yak-3s, while the South Koreans sported M1 Garands, M4 Shermans, and P-51 Mustangs (both armies largely reserved their most advanced, new weapons for each other). South Korean forces also had a large advantage in artillery and naval forces, though the two armies were equally matched in armored and aerial capabilities.[2]
However, Lee interfered in the military plans. The original plan for the invasion of the North was simply a quick strike to blitz the numerically inferior North Korean army on the border and attempt to encircle and destroy the bulk of North Korean forces before they could retreat to Pyongyang and mount an urban defense of the city, akin to Stalingrad (1941), Berlin (1945), or Zagreb (1955). The goal of this plan was to capture Pyongyang relatively unharmed and with low levels of civilian casualties. Lee vetoed this plan. He insisted that victory over the North would be pointless if Communism wasn't truly crushed, if the North Koreans could escape into North China or the Soviet Union. Instead, Lee ordered his general staff towards an alternative plan. In Operation Pukchin Tongil, the South Korean Army was to launch a grand offensive across the border. However, this was only to be a distraction - the real objective was to ship two armies across the sides of Korea, landing in Yongyu (to the North of Pyongyang), erect large defensive perimeters to the North, and then advance south from there. The landings would be aided by disguised American merchant marine ships - South Korea had enough warships, but not enough transports, and in one of the many controversial acts of his presidency, Russell ordered US Merchant Marines to disguise as ROK Navy ships to help. With the bulk of the South Korean Army advancing from the South, they would pincer both the North Korean Army as well as the DPRK government in Pyongyang, turning it into a grand battle of annihilation.
By and large, the North Korean Army took the bait, rushing troops down to the border as South Korean artillery and infantry crashed against North Korean positions. The initial "victories" of the North Korean Army lured Kim il-Sung into complacency, a complacency brutally crashed by landings in relatively undefended Kaesong and Wonsan. Within a week, the bulk of North Korean forces were being pushed from both south and north into Pyongyang, where they rapidly prepared to make their last stand. The North Korean government, defiant, took inspiration from the Battle of Moscow, when Stalin refused to flee Moscow, assuming that they could easily hold out long enough for Soviet intervention. This proved to be a grievous mistake. The biggest advantage of the North Korean Army was its parity in armored and aerial power. Stuck in a city, their tanks were less effective than otherwise and any airfields set up were under constant artillery fire from South Korean forces.
Like Yugoslavia, losses in Pyongyang were horrific, but they were exceptionally horrific due to the policies of both governments. Lee intended on Pyongyang being the grave of most of North Korea's Communist Party, including its leadership, and he ordered the army to target any group of individuals fleeing the city. Lee feared that North Korea's leaders could sneak out in disguise - as a result, caravans of refugees were typically arrested and interned to help sift out government members - however any caravans that couldn't be stopped were bombed and machine-gunned by South Korean forces, This led to more bloodshed, as this rapidly stiffened the resolve of the North Korean Army, which was then actually able to conscript many of those civilians that would have otherwise escaped. Under orders from the DPRK government, civilians were forcibly drafted and commissars roamed the streets of Pyongyang, executing those who fled or deserted from the widely unpopular conscription orders.
This also led to a diplomatic incident that embarrassed the US - one Merchant Marine captain, Leonard LaRue of the USS Meredith Victory, abandoned his mission (dropping off equipment to South Korean marines) to load almost 20,000 civilians trying to escape from Pyongyang. Outraged South Korean planes bombed the ship, trying to sink it. Miraculously, the ship escaped sinking - LaRue said while he credited Jesus Christ for protecting him, he also pointed out that he suspected the individual pilots were intentionally missing his ship (the theological argument was naturally that Christ was working through them). When news of the incident leaked, the American press largely expressed anger that the US Navy did little to nothing to protect the ship - even though public sentiment was widely in favor of the war and of American support of South Korea, the story made many Americans concerned about civilian casualties.
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[1] I recently visited the Korean War museum in Seoul and they had a pretty good diagram on the order of battle on the eve of the war in 1950. ITL, the North/South ratio of troops was 198k/106k, with NK advantages in tanks and planes (though parity in ships/artillery). However, 1/3rd of the North Korean troops (and generally the crack troops) were veterans of the People's Liberation Army, lent to Kim Il-Sung by Mao. ITL, the PRC doesn't really have that kind of luxury, so the KPA is a lot smaller and more poorly trained (and gets overrun easily).