Dual monarchy occurs when two separate kingdoms are ruled by the same monarch, follow the same foreign policy, exist in a customs union with each other and have a combined military but are otherwise self-governing. The term is typically used to refer to Austria–Hungary, a dual monarchy that existed from 1867 to 1918.
In the 1870s, using the Dual Monarchy of Austria–Hungary as a model, the Prince of Wales (later King Edward VII) and William Ewart Gladstone proposed that Ireland and Great Britain form a dual monarchy. Their efforts were unsuccessful, but the idea was later used in 1904 by Arthur Griffith in his seminal work, The Resurrection of Hungary. Griffith noted how in 1867 Hungary went from being part of the Austrian Empire to a separate co-equal kingdom in Austria-Hungary. Though not a monarchist himself, Griffith advocated such an approach for the Anglo-Irish relationship. The idea was not embraced by other Irish political leaders, and Ireland eventually fought a war of independence (1919–1921) to leave the Union of Great Britain and Ireland and form a separate state, the Irish Free State in 1922.
Later historians have used the term to refer to other examples where one king ruled two states, such as Henry V and Henry VI, who were effectively kings of both England and France in the fifteenth century as a result of the formation of a puppet state in a large area of France during the Hundred Years' War, Denmark–Norway, a dual monarchy that existed from 1536 to 1814, the Iberian Union between Portugal and Spain (1580-1640), and the Polish-Lithuanian Commonwealth (1569-1795).
A dual monarchy is not necessarily a personal union. In a personal union two or more kingdoms are ruled by the same person but there are no other shared government structures. States in personal union with each other have separate militaries, separate foreign policies and separate customs duties. In this sense Austria–Hungary was not a mere personal union, as both states shared a cabinet that governed foreign policy, the Army and common finances.