The Glossary of Linguistics Terms at the SIL website characterizes clause-chaining by "the possibility of long sequences of foreground clauses with operator dependence."
In typical clause-chaining, a clause with a fully inflected finite verb can lead into or follow a string of clauses whose verbs are only partially inflected, which verbs are called "medial verbs." These strings of clauses may represent events in temporal order (as if "then" were a medial verb suffix), causal order (as if "which caused" were a medial verb suffix) or other kinds of sequences.
The languages of Papua New Guinea are famous for their various clause-chaining schemes, which are often described as alien to Indo-European languages.
Languages with clause-chaining such as we find in the Papua-New Guinea languages often (but not always) have markers for "switch reference," i.e. whether the subject of the clause is the same or different than the subject of the previous clause.
My question is, how do I interpret the above definition of clause-chaining provided by the SIL? What is a "foreground clause"? What is "operator dependence"? Can you give or cite examples these concepts, and of clause-chaining?
I would be particularly interested in sentences written in a hypothetical form of English that had a simple form of clause-chaining. I want to get a feel for how clause-chaining is used.