The tape, supposedly a telephone recording of then US president Reagan and the British prime minister, was anonymously sent to several Dutch newspapers during the 1983 UK general election.
In the faked exchange about the 1982 Falklands War between Britain and Argentina, Reagan urged restraint while Thatcher wanted Buenos Aires punished "as quickly as possible".
The tape was constructed by cutting up and reassembling sound clips of the two leaders.
The British punk-rock band Crass admitted making the tape - but that was only after it was taken seriously enough that the CIA and Britain's MI6 had sought to establish whether the Soviet Union or Argentina was behind it.
It said the British embassy in The Hague had passed the tape to London.
"This looks like a rather clumsy operation. We have no evidence so far about who is responsible," it said.
"SIS doubt whether this is a Soviet operation. It is possible that one of the Argentine intelligence services might have been behind it; or alternatively it might be the work of left-wing groups in this country."
The papers have been declassified after 30 years in the National Archives.
