The 1921 Anglo-Irish treaty was an attempt to bridge the Irish desires for an independent ‘Republic of Ireland’ with the British requirements for imperial unity and defence with Ireland remaining under the crown. While it did go a step beyond the 1912 Home Rule Act, it didn’t go far enough to prevent a split within Sinn Féin between the anti-treaty faction led by Éamon De Valera, who would only accept a fully independent republic, and the pro-treaty faction led by Michael Collins, who would famously describe it as “not the ultimate freedom that all nations aspire and develop to but the freedom to achieve it”.

The ratification of the treaty in January 1922 saw the anti-treaty ministers leave the Provisional Government and the anti-treaty IRA convene to determine their policy towards the Free State and the treaty. Collins and De Valera would attempt to negotiate and avoid a civil war, however the seizure of the Four Courts in Dublin in April by anti-treaty forces, and the assassination of Field Marshal Sir Henry Wilson in London in June, led the British government to issue an ultimatum for the Provisional Government to re-capture the Four Courts. Fearing a British reoccupation of the country, the National Army assaulted the Four Courts and began the Civil War. During this conflict Michael Collins would be ambushed and killed on 22 August 1922 (specifically mentioned because it was recently the 100-year anniversary).

While the British government had funded and armed the National Army at its inception, they also provided considerable material support to the Provisional Government during the Civil War, it was even offered by Churchill to provide British pilots and planes in Free State colours. It was this material support that would give the Provisional Government the edge needed to win the Civil War and form an elected government under Cumann na nGaedheal.

Post-war the Free State would remain economically tied to the United Kingdom, free trade continued, the Irish pound was pegged to the sterling, and when Britain abandoned the gold standard in 1931 the Free State soon followed. The Free State attempted to complete the work of the pre-independence Land Acts in 1932 (a number of acts that sought to give Irish tenant farmers ownership of the land) which was only possible with a British guarantee on a £30 million loan.

The 1932 election saw De Valera returned to the reigns of power at the head of his Fianna Fáil party, founded due to reluctance of Sinn Féin to acknowledge the Free State. De Valera rejected the laissez-faire policies of the preceding government and sought to become self-sufficient and to dismantle the political and economic ties with Britain.

Between 1932 and 1937, De Valera removed the right of appeal from the Irish courts to London, abolished the Oath of Allegiance (to the monarch), removed all references to the King in the Irish constitution, removed the Governor-General who gave the crown’s assent to Irish legislation, and finally introduced a new constitution in 1937 which proclaimed the State of Eire.

De Valera refused to pay the land annuities, the debt repayment from the earlier Land Acts which commenced the Anglo-Irish trade war. This saw Britain introduce a 20% tariff on Irish agricultural goods, and Ireland a retaliatory 20% tariff on British coal and sugar. The economic war of back-and-forth tariffs would prove much more damaging to the Irish economy than the British as 90% of Irish exports were to the UK while 7.8% of British exports were to Ireland.

There was a lull in 1935 when the two governments entered a Coal-Cattle pact in which Irish cattle quotas were increased by 33.3% in return for a monopoly of Irish coal imports to the UK, however the true end came in 1938. Both countries were eager to end the war as Britain turned its attention to a rising Germany and Fianna Fáil were returned with a slimmer majority in the 1937 election.

The subsequent Anglo-Irish Agreement was highly favourable to the Irish; the 1921 “Treaty ports” of Berehaven, Spike Island, and Lough Swilly were to be returned from the Royal Navy to Irish ownership, land annuities were to be settled with a lump sum payment of £10 million, and British tariffs were to be removed but only a reduction of Irish tariffs.

I feel the answer linked by u/Georgy_K_Zhukov sufficiently covers the period of World War 2, or “The Emergency” as it was called in Ireland, but I will add where you may have gotten 1946 from (1948 really but read on).

A declaration of a republic had long been a goal of the Irish state, upon coming to power De Valera had essentially created one with the 1937 constitution but refused to apply the name while the island remained partitioned. To add to the ambiguity, during the 1936 abdication crisis of Edward VII, De Valera rushed through the External Relations Act which removed the monarch from internal affairs but maintained him as external symbol of co-operation among the Commonwealth. Fianna Fáil lost the 1948 election to a coalition of parties led by John A. Costello. While on a visit to Canada he suddenly announced they would repeal the External Relations Act and remove Ireland from the Commonwealth, which was followed by the Republic of Ireland Act 1948.

Sources:

Various, Atlas of the Irish Revolution, Cork University Press, 2017

S. J. Connolly, Companion to Irish History, Oxford University Press, 2011

T. Pusova, “Irish Economic Policy in the 1930s and the Authority of Eamon de Valera”, The Journal of European Economic History, 2019

A. W. Bromage, “Anglo-Irish Accord”, Political Science Quarterly, 1938

I. McCabe, “John Costello 'Announces' the Repeal of the External Relations Act”, Irish Studies in International Affairs, 1992

More can always be said, but this older answer might be of interest for you, although it is only focused on WWII.