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.
For class forcing I'd suggest to start with Sy Friedman's work which can be found here.
In particular, his chapter from the Handbook of Set Theory (available on the above site) can be used as a good start.
The problem with class forcing is that classes are not "real" objects in the universe of set theory. They are formulas interpreted in the model as definable collections. This means that arguments which you can get "for free" from sets are now very expensive in the sense that you need to verify things. Global choice makes things easier because it keeps all classes in the same size and allows us to choose from everything at once.