The terms reper or reper bundle appear a few times in Bakalov/Kirillov's "Lectures on Tensor Categories and Modular Functors", and I vaguely recall having encountered them in a course on gauge field theory. But I have since forgotten their meaning.
Googling "reper bundle" does not yield anything, and no question about these has been asked. I seem to remember that it was some kind of frame or frame bundle?
I believe a reper bundle is a frame bundle. There are several sources defining a "reper", e.g. here:
A local reper or frame of $E$ over $U \subseteq M$ is a collection of sections $e_1, . . . ,e_k$ such that for all $p\in U$ the vectors $e_1(p),\ldots ,e_k(p)$ form a basis of $E_p$. A reper is called global if this property extends to all $p\in M$.