According to the useful references cited by Martin, tutela de evidência is a form of urgent summary relief, similar to an injunction, which is available when the evidence in support of the claim is incontestable. Martin's text says it is similar to a mandado de segurança except that it is only available against private parties and not the state. i.e. "ação intentada pela pessoa no sentido de ser assegurado um direito certo e incontestável, ameaçado ou violado" (...) "Dos writs que mais se assemelha ao mandado de segurança criado pelo Direito Brasileiro é o writ of injunction" Vocabulário Jurídico - De Plácido e Silva.

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Note added at 1 day44 mins (2012-05-12 23:04:07 GMT)
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I refer only to Martin's first reference as I do not think the second one is relevant. Evidência here does not mean evidence (false friend) but means that the applicant's case is unequivocal.