Finding the "complexification" of a real vector space is trivial; essentially the same as obtaining the complex numbers from the real numbers. However, when complexifying a real Banach space, there are a range of choices, for a norm that fits in naturally with the norm of the real Banach space. In this talk it is proved that one of these choices has a property that one expects, namely that the complexification of the projective tensor product of any two real Banach spaces is isometrically isomorphic to the projective tensor product of their complexifications.