Mark Ellison QC, prosecuting, told the court that it had been a "classic example of blackmail".
The two men were arrested in a sting operation at a London hotel by undercover police officers last September.
The Old Bailey heard that Strachan was a "Walter Mitty"-type fantasist who routinely claimed to be a friend of royals and lived an extravagant lifestyle.
McGuigan, who is a recovering alcoholic, contacted A's representatives last July and was present as the tapes were played to an undercover officer, who posed as a royal aide, in a meeting shortly before the pair were arrested.
The judge, Mr Justice Cooke, said the offence was so serious that the pair had to be jailed.
He told them: "The corrosive evil of blackmail means that any sentence must have a deterrent effect."
And the judge said he was sure that the men had acted together to target the royal - a victim who was "particularly susceptible".
Reconstructed excerpt of blackmail recording
Mr Justice Cooke said the two men had encouraged D to make "scurrilous accusations" and collected "scurrilous and salacious" material against A and his family.
Following the guilty verdict, Crown Prosecution Service (CPS) reviewing lawyer Mark Carroll said: "Although they claimed to be acting in the interests of the victim, the jury rejected this story and agreed with the prosecution that they were simply interested in the money.
"Prosecutions for blackmail are rare but it is a serious offence which can involve the victims making themselves vulnerable in order that the case comes to trial."
He added that the CPS "will not hesitate to prosecute blackmail cases and we will always seek to protect the anonymity of blackmail victims".
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