Question: What Islamic state controlled southern Iberia until 1031?
Sentence: Southern Iberia remained under control of Islamic states, initially under the Caliphate of Córdoba, which broke up in 1031 into a shifting number of petty states known as taifas, who fought with the Christians until the Almohad Caliphate re-established centralised rule over Southern Iberia in the 1170s.
Answer in sentence: yes