Muslim World including the Islamic Republic of Pakistan, Iran, Afghanistan, and Mauritania. Pakistan adopted the title under the constitution of 1956, iran adopted it after the 1979 Islamic Revolution that over throw the Pahlavi Dynasty. Afghanistan adopted it after the 2001 overthrew of the Taliban. Despite the similar name the countries differ differ greatly in their Govts and laws.
The term "Islamic republic" has come to mean several different things , some contradictory to others. To those Muslim Religious leaders in the middle east and Farca who advocate it, an Islamic republic is astate under particular theocratic form of govt. They see it as a compromise between a purely Islamic Caliphate , and secular nationalism and republicanism . In their conception of the Islamic Republic Republic, the penal code of the state is required to be compatible with some laws of of Sharia , and the state may not be a monarchy as many middle eastern states are presently.
Pakistan was the first country to adopt the adjective "Islamic" to modify its republican status under the otherwise secular constitution of 1956. Despite this definition the country did not have a state religion until 1973, when a new constitution , more democratic but less secular , was adopted. Pakistan only uses the Islamic name on its passport. The constitution of Pakistan , part IX, article 227 says :all existing laws shall be brought in confirmitywith the injunctions of Islamas laid down in the Quran Sunnah, in this part referred to as the injunctions of Islam, and no law shall be enacted which is repugnantto such injunctions.