Let $G$ be group (for simplicity -- abelian). The isomorphism classes of $G$-principal bundles over $X$ are classified by first cohomologies of certain sheaf on $X$ (e.g. $H^1(X, G)$, if $G$ is discrete, or $H^1(\mathcal{O}_X)$ for holomorphic $U(1)$-bundles over complex manifolds etc.)
As one can take sum of two cocycles, it gives us a structure of abelian group on the set of isomorphism classes of $G$-bundles. How to describe it explicitly? For example in case of $U(1)$-bundles it corresponds to tensor product of two line bundles. And what to do in more general case?
I guess, that one must take fibered product of two principal bundles $P_1 \times_X P_2$ and then take quotient over relation $(gp_1, p_2) \sim (p_1, gp_2)$. But I'm not sure if the construction is correct, and I would love to have some references in any case.