When Jech, in his Set Theory, deals with forcing with a class of forcing conditions (with the aim of proving Easton's theorem), he starts with the assumption that there is a well-ordering of the ground model, i.e. the ground model satisfies the Axiom of Global Choice.
Having examined and, to a degree, understood the development in this section, I can't figure out where he actually uses this assumption. The only place I can see where this might be relevant is in the discussion on the existence of a generic set and even here it seems Global Choice isn't needed in every possible solution. For example, if you justify forcing via a reflection theorem argument or a countable ground model, I don't believe you need Global Choice.
On the other hand, you can take the Boolean-valued semantics approach and define the canonical name for the generic set as $\dot{G}(\check{p})=p$ for a forcing condition $p$ (assume here that the forcing notion is a proper class Boolean algebra). So far we're fine, $\dot{G}$ is a class in the Boolean-valued model, everything is rosy. Conceivably, if we were to define a class $\check{M}$, representing the ground model in the Boolean-valued model, via $\|x\in\check{M}\|=\bigvee_{y\in M}\|x=\check{y}\|$ the Boolean-valued model would see itself as the generic extension of $\check{M}[\dot{G}]$, since this holds when forcing with a set of conditions. Of course, there is a problem in defining $\check{M}$ this way, since we can't generally take sups of a proper class of (different) Boolean values.
I expect this approach should be salvageable, using Global Choice. In particular, I think we should be able to take the offending sup along the given well-ordering of $M$ and somehow "stagger" it, so it becomes well defined.
I'm not at all sure if this is legitimate or if it even leads anywhere, so I would appreciate comments and an explanation of what is really going on. Additionally, can anyone suggest another reference for class forcing? I generally enjoy Jech's book, but I found this section to be somewhat opaque and hard to understand.