Here's a few ideas. Have the civil war go considerably better for the Reds. In addition the Reds maintain their alliance with the Black Army and when the Kronstadt uprising happens the leadership of the Communist Party decide to negotiate instead of violently suppress.
As a result of their stronger position, when the Red Army pushes the Japanese out of Vladivostok they decide to continue fighting and liberate Korea. Fearful of the growing influence of the anarchists, decide to put Makhno and the Black Army in charge of the operation so that they'll be far away. The liberation is successful, and the combination of the largely anarchist liberation force, combined with a strong local anarchist movement, means that the political situation in Korea is heavily dominated by anarchists.
Back in Soviet Ukraine the Soviet forces begin a crackdown on the Ukrainian Free Territory. Resistance is much stiffer than expected, and the Korean Free Territory threaten to attack Eastern Siberia in retaliation. Eventually cooler heads prevail and a compromise is met whereby the Soviets recognise the anarchist order in Korea and will agree to uphold it as an experiment in proletarian organisation (IOTL there were some in the Communist Party who argued in favour of using the Free Territory as an experiment), in exchange for the subordination of the Ukrainian Free Territory to Soviet control. This results in an uneasy peace between the Blacks and the Reds, whilst also giving the Soviets a useful dumping ground for the more hard-line anarchist militants.
In China anarchist ideas, instead of Communism, become the hegemonic ideology of the far-left (particularly in the trade unions many of which affiliate to the IWW), in part due to the Korean anarchists being seen as more "Asian" than the "European" Soviets, and the far-left as a whole is in a stronger position due to Soviet and Anarkorean support for the KMT during the Northern Expedition. Chaing Kai-shek attempts a coup to eliminate the leftist influence in the KMT, which triggers a civil war between the rightist Chaing and an anarchist led alliance of Anarchists, Communists and fellow travellers. The Soviets and Anarkoreans intervene on the side of the leftists, whilst the Japanese, desperate to regain influence and prestige following their expulsion from Korea, intervene on the side of the rightists. Whilst the rightists are initially successful, managing to take the major cities and the Japanese landing forces in southern Korea and Eastern Siberia, the leftists eventually win out through a protracted guerilla struggle and peoples war in the countryside (whilst also being able to use Soviet occupied regions in Manchuria and northern China as bases for resupplying and reorganising).
At this point multiple military defeats had destroyed the myth of Japanese invulnerability, the armed forces have been severely beaten and a filled with dissidents, the officer corps has been decimated through a series of fratricidal coups and counter-coups, and the success of anarchism in Korea and China has ensured that the Anarcho-Syndicalist current in the Japanese left remains strong instead of being subsumed under Communism as IOTL. After Hirohito is assassinated by anarchist terrorists Japan falls into a brief civil war, with the anarchists coming out on top.
Rising tensions between the Soviets and their anarchist allies, combined with the growing power of the nascent anarchist bloc in Asia eventually results in a split between the two forces.