The following definition is in Ahlfors (1979), but also found in Weyl or perhaps Weierstrass.
Two function elements (f1, U1) and (f2, U2) are said to be analytic continuations of one another if U1 ∩ U2 ≠ ∅ and f1 = f2 on this intersection.
A global analytic function is a family f of function elements such that, for any (f,U) and (g,V) belonging to f, there is a chain of analytic continuations in f beginning at (f,U) and finishing at (g,V).
Using ideas from sheaf theory, the definition can be streamlined.
In these terms, a complete global analytic function is a path-connected sheaf of germs of analytic functions which is maximal in the sense that it is not contained (as an etale space) within any other path connected sheaf of germs of analytic functions.