NP-complete variants of NPI problems Motivated by these posts, An NP-complete variant of factoring and Relationship between symmetry and computational intractability, It seems to be worthwhile to investigate the different factors that increase the hardness of problems in $NPI$ from an intermediate complexity to full $NP$-completeness. I'm interested in other $NP$-complete variants of $NPI$ problems. Ideally, a recent survey paper of $NP$-complete variants of $NPI$ problems would be the best answer?

EDIT July 1st, 2012 The bounty will be awarded to the person that lists the maximum number of such problems.
From Papadimitriou's paper "On total functions, existence, theorems and computational complexity":

Nondeterministic multivalued functions with values that 
are polynomially verifiable and guranteed to exist form an
interesting class (called $TFNP$) between P and NP ...

.... It is quite interesting, in view of Theorem 2.1,
that in some cases of problems in $TFNP$, if together with the input
we are also given one of the solutions
that are guaranteed to exist, then the problem indeed becomes NP-complete ...

For example the TRICHROMATIC TRIANGLE problem is in $TFNP$, but the SECOND TRICHROMATIC TRIANGLE is $NPC$ (the paper contains a sketch of the proof).