I was always taught that if you can replace a participle with a pronoun, then you should use the possessive case. So instead of "I saw him eating", because I can substitute "it" and say "I saw it", I should say "I saw his eating." "His eating" is the object here, and it can be moved around as such. For example, I might say "His eating is good," but it would be incorrect to say "Him eating is good." This rule of "Can I substitute ‘it’?" has proven helpful in most cases.
That said, when I come to very long participial phrases, using the possessive can seem awkward. For example, even "I saw him eating a donut" sounds much more natural than "I saw his eating a donut", even though I can still substitute it for the entire phrase and say "I saw it." But if "I saw him eating a donut out on the front lawn yesterday morning with a friend," then it seems even more awkward to use the possessive, even though it still seems more grammatically correct.
Any thoughts on this? What is the rule? When is it proper to use a possessive in combination with a participial phrase, such that the entire thing acts as an object, and when is it proper to use the phrase to modify the object?
After thinking about this a bit, I’m thinking that maybe it just depends on the focus of the statement. If the action of a person’s eating is my focus, then I would say, "I saw his eating." But if the person is my focus and he also happens to be eating, I would say, "I saw him eating."
The rule here seems to be, "If I can remove the participial phrase and still have the statement make sense, then I should use the objective instead of the possessive." If my focus is "him", then "eating" is just extra info and can be removed, so "I see him eating" would be correct. But if my focus is "eating," then removing it would make the sentence nonsensical, so "I see his eating" would be correct.
So the final answer is, both could be correct, and which should be used depends on context.