NP-complete variants of NPI 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).