Both Sovereign Base Areas currently claim 3 nautical miles as their own exclusive economic zone, but reserve the right to claim up to 12.
According to our research, this was defined in the treaty concerning the establishment of the Republic of Cyprus, 19 August 1960. (source)
Territorial waters of three nautical miles are claimed, and the right according to the laws of the United Nations to extend the claim of up to twelve nautical miles is reserved. The UK has a potential claim to an exclusive economic zone out to 200 miles (322 kilometres) (that could include the Aphrodite gas field), as the treaty establishing the Republic of Cyprus (Annex I, Section III) specifically excluded any Cypriot claim to the two maritime areas adjacent to the bases.
An interesting* twist to the already complicated situation regarding the eventual gas-extraction process.
* For lack of a better word.