How were the prime ministers of Japan selected during the Taisho era?

Under the Meiji Constitution of Imperial Japan, the Prime Minister - like all Ministers of State - were technically appointed by the Emperor solely on his own discretion.

In practice, appointments were always made on the recommendations of the genrō, a clique of prominent elder statesmen who had orchestrated the Meiji Revolution. Whenever a vacancy arose, the genrō would meet and name a replacement. If the nominee accepted, the Emperor would then formally appoint him as the next Prime Minister. In the last years of the Meiji Era, the Genro typically accepted the recommendations of the outgoing Prime Minister.

The Meiji founders bequeathed a system in which a group of Elder Statesmen or Genro advised the throne regarding the appointment of the Prime Ministers ... the Emperor (who had no personal say in the matter) formally appointed the new Prime Minister.

_{- Kasza, Gregory James. The State and the Mass Media in Japan, 1918-1945. University of California Press, 1988.}

Up until the late Meiji and sporadically into the early Taishō period, there existed in Japanese politics a concept that the government should be "above" party politics. This doctrine, known as chōzen shugi, facilitated the appointment of bureaucrats or nobles to the premiership by the elders, without due regard to the balance of power within the Diet. An example is the genrō Yamagata Aritomo, the 3rd and 9th Prime Minister of Japan.

Yamagata's Cabinet represented a continuity with precious Cabinets in philosophy as well as personnel. He fully subscribed to the concept of choen shugi (the principle of aloofness), which meant in practice that the governmnet should remain above and aloof from political parties.

_{- Hackett, Roger F. Yamagata Aritomo in the rise of modern Japan, 1838-1922. Cambridge, Mass: Harvard University Press, 1971.}

Such appointments were already increasingly unpopular towards the end of Meiji, and went into crisis in the first year of Taishō (1912). When the genrō nominated Katsura Taro as the next Prime Minister, they provoked a massive backlash in the Diet that forced Katsura to resign. Despite this, the genrō retained control of the Prime Ministerial appointment process and continued to allow their personal distaste for party politics or specific politicians continued to factor into their decision making process. At the same time, however, The elders became more sensitive to the need for support within the Diet.

The Diet then cast a no-confidence vote against [Katsura], who could have dissolved the House of Representatives and called for new elections. He chose to resign instead. This series of events marked a turning point in Taisho politics. The political parties in the Diet and the people had demonstrated that no prime minister or cabinet could be selected without regard for public sentiment. No longer could the genro ignore the political makeup of the Diet when choosing cabinets.

_{- Menton, Linda K. The Rise of Modern Japan. University of Hawaii Press, 2003.}

One major milestone was reached on 1924, when the electoral landslide of the Constitutionalist parties caused the elders to nominate Katō Takaaki. This established a customary rule the leader of the biggest party in the Diet to be appointed Prime Minister. However the genrō continued to formally choose the a candidate, even though the death of Matsukata Masayoshi later that year left Saionji the sole remaining elder.

[Hara Kei] and his supporters were not claiming that the emperor could not continue to appoint cabinets and prime ministers populated by the genro; what they were asserting is that the emperor had the option of not relying upon a genro oligarchy, rather choosing to appoint party cabinets along British lines if he so chose.

_{- Mosk, Carl. Japanese Economic Development: Markets, Norms, Structures. Routledge, 2007.}

Though the system had largely fallen apart with the death of most of the genrō, it nonetheless it lasted until the rise of militarism in early Shōwa. In 1937 the last genrō resigned his powers and saw them replaced by a similar but less antiquated group of elder statesmen. Thereafter the premiership became dominated by the military.

The formation of the Hayashi Cabinet in January 1937 was the last time that the Genro took responsibility for answering the throne on a cabinet change ... Saionji was persuaded to retain the title of Genro but insisted that a new procedure be drawn up transferring to the Lord Keeper of the Privy Seal the responsibility for recommending cabinets hitherto held by the Genro ... The new system [was] finalised in May 1937.

_{Connors, Lesley. The Emperor's Adviser: Saionji Kinmochi and Pre-war Japanese Politics. Routledge, 2010.}