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$NP \cap coNP$ is not known to have complete languages, but I don't know of any consequences as strong as what Marcos claims. Juris Hartmanis and his students worked extensively on this problem in the early 80's. Two references I know are:

Michael Sipser: On Relativization and the Existence of Complete Sets. ICALP 1982: 523-531

This paper shows that there is an oracle relative to which $NP \cap coNP$ does not have complete sets. So proving that it does have complete sets would at least require non-relativizing techniques. Also, as mentioned by Peter in Marcos' comment, there's also Hartmanis and Immerman's work:

Juris Hartmanis, Neil Immerman: On Complete Problems for NP$\cap$CoNP. ICALP 1985: 250-259

They give several interesting structural results about the problem. For one, they complement Sipser's result, giving an oracle relative to which $NP \cap coNP$ has complete languages yet $P \neq NP \cap coNP \neq NP$. They also show that $NP \cap coNP$ has complete languages under many-one reductions iff it has complete languages under Turing reductions. This paper also cites the following neat characterization by Kowalczyk (1985): $NP \cap coNP$ has a complete language iff there is a recursively enumerable list of pairs of $NP$ machines {$(N_{i,1}, N_{i,2})$} such that $\overline{L(N_{i,1})} = L(N_{i,2})$ and $\bigcup_i L(N_i) = NP \cap coNP$.

But since then, there hasn't been much progress on the question, to my knowledge. I'd be very happy if I were corrected...

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$NP \cap coNP$ is not known to have complete languages, but I don't know of any consequences as strong as what Marcos claims. Juris Hartmanis and his students worked extensively on this problem in the early 80's. Two references I know are:

Michael Sipser: On Relativization and the Existence of Complete Sets. ICALP 1982: 523-531

This paper shows that there is an oracle relative to which $NP \cap coNP$ does not have complete sets. So proving that it does have complete sets would at least require non-relativizing techniques. Also, as mentioned by Peter in Marcos' comment, there's also Hartmanis and Immerman's work:

Juris Hartmanis, Neil Immerman: On Complete Problems for NP$\cap$CoNP. ICALP 1985: 250-259

They give several interesting structural results about the problem. For one, they complement Sipser's result, giving an oracle relative to which $NP \cap coNP$ has complete languages yet $P \neq NP \cap coNP \neq NP$. They also show that $NP \cap coNP$ has complete languages under many-one reductions iff it has complete languages under Turing reductions. But since then, there hasn't been much progress on the question, to my knowledge. I'd be very happy if I were corrected...