Wikipedia: "Antiparticles have exactly opposite additive quantum numbers from particles, so the sums of all quantum numbers of the original pair are zero."
Is it possible to annihilate a quark-antiquark pair if the antiquark has a color different from the quark?
Eg.: Will red up and anti-green anti-up annihilate? And if so, what is the product?
Usually the word "annihilation" is reserved for interactions whose inputs are two particles that are each other's antiparticle, and your example is not annihilation by that definition. But you can have an interaction where two different quarks go in and something else comes out, as ACuriousMind already mentioned. We happen to have a special word for annihilations, but they are governed by the same physical laws as any other interaction.
Well, you can just take the ordinary quark/anti-quark/gluon vertex for that and get a red/anti-green gluon (which will then decay/hadronize further, since it is color-charged and thus confined).