Rome (CNN) The Vatican has ordered two of its own tombs to be searched -- the latest twist in the mysterious disappearance of a teenager, 36 years ago.

On Tuesday, Gian Piero Milano, the Vatican's Promoter of Justice, authorized two exhumations in response to a petition launched by the teenager's family, who believe that her body is buried at the Teutonic Cemetery in Vatican City.

Orlandi's mother and brother still live inside the Vatican's walls and have continued to push for her case to be investigated throughout the years. The family's lawyer received an anonymous tip last summer suggesting that they should "seek where the angel indicates," Italian news agency ANSA reported

The clue led the family to a wall in the Teutonic cemetery which features an angel pointing. The tombs will be opened on Thursday July 11, Vatican spokesman Alessandro Gisotti announced Tuesday.

Gisotti described the case as "a long and painful one" but also appeared to try to temper expectations of a quick resolution, adding that any bones found during the search would have to undergo laborious DNA analysis.

Demonstrators gather around Orlandi's brother Pietro, center, during a event marking the 30th anniversary of her disappearance in St Peter's Square in 2013.

Gisotti said the operation would be a "complex organization of personnel and machinery" involving the Vatican Gendarmerie in addition to Vatican construction workers. He also said the Vatican probe would only handle the part concerning territory in the Holy See. Over the years, the wider investigation into Orlandi's disappearance has been handled by Italian authorities since she vanished in Italian territory.

The Teutonic Cemetery is adjacent to Saint Peter's Basilica. It contains the remains of the Swiss Guards who died defending the Vatican City against forces of the new Kingdom of Italy in 1870. It is also the reserved burial ground for members of German-speaking religious institutions in Rome.

The Orlandi case has enthralled Italians and sparked numerous conspiracy theories, which have pointed the finger at everyone from the Mafia to the Vatican itself.

In 2012, a top Vatican exorcist, the Rev. Gabriele Amorth, told CNN he suspected the young girl had been abducted for sexual reasons, adding: "The investigation should be carried out inside the Vatican and not outside."

Bone fragments found during construction work at the Vatican Nunciature Embassy last year reignited speculation over Orlandi's disapperance.

In 2005, an Italian detective received an anonymous tip off that Orlandi had been kidnapped by order of then-vicar of Rome, Cardinal Ugo Poletti, and that "the secret to the mystery lies in a tomb in Sant'Apollinare basilica" -- specifically the crypt that holds mob boss Enrico "Renatino" De Pedis.

De Pedis was gunned down in Rome in 1990 and his body was moved to the basilica sometime before 1997.

It was not the first time De Pedis' name had been linked to the Orlandi case. In 2008, his mistress alleged that De Pedis had been involved in the teen's kidnapping and that her body had been buried in the foundations of a property on the outskirts of Rome.

Investigators followed up on the claims but found that the building's concrete foundations had been poured before the girl disappeared.

De Pedis' tomb was searched in 2012 but provided no further developments.