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Disjoint NP-pairs are an interesting model of computation with important applications in cryptography and proof complexity. The question whether there exists a complete disjoint NP-pair was posed by Razborov in 1994 and is one of the most important problems in the field. In this paper we prove that there exists a complete disjoint NP-pair which is computed with access to a very weak oracle (a tally NP-oracle). In addition, we exhibit candidates for complete NP-pairs and apply our results to adoi:10.1109/synasc.2009.9 dblp:conf/synasc/Beyersdorff09 fatcat:tpujekk3uvgytiwd4q6ekudd5i