There are two different mechanisms at play here, I'm not sure which you are most interested in but we can define them and then perhaps progress from there.

First we have what some historians would term the Angevin Empire. This was the realm presided over by Henry II and his sons Richard the Lionheart and John. The term 'empire' to describe this territory comes in and out of fashion, but that is a discussion for elsewhere.

These kings were very powerful, because they controlled such a large realm, but it was not a single state. For example, Richard (1189-99) ruled not as a emperor of this region as we might understand it, but individually and distinctly in each. He was king of England, duke of Normandy, duke of Gascony, duke of Aquitaine, etc. These rights and titles had been accumulated over generations by military conquest and marriage (particularly that of his parents). Likewise, when Henry subjugated Ireland, it was still ruled as a lordship.

The concept of a monolithic and unified France did not exist in this period as it does now. Its geographical regions were divided considerably by culture, language and allegiance. Lords would still technically owe homage to the crown, but their levels of independence and separation could be extreme.

This meant that the regions 'controlled' by the Angevin kings of England shared the same ruler but were not incorporated into a single state. Each retained their own unique cultural identity and hierarchies. England, while the most prestigious of their titles, was still one of many. For example, Richard was king for a decade but only spent perhaps nine months there.

The other and somewhat distinct concept of an Anglo-French union was witnessed during the Hundred Years War. During this period (1337-1453), the kings of England did not assert control in France as lords and dukes under the French king, but rather asserted that they were the legitimate kings of France themselves.

It was this contest between the crowns that lead to many famous battles, including the triumph of Henry V at Agincourt, 600 years ago today. His son, Henry VI, therefore became king of England in August 1422 and was crowned king of France in Paris three months later (all before his first birthday). In this way, the English kings ascended from the status of over-mighty vassals in France to pretensions of absolute dominance. In reality, the English and their allies ruled in the North, while the French ruled in the South.

Hopefully this sheds a little light on the two distinct phases of this relationship, hopefully it was one of these which your question referred two. Obviously, both are massive topics, but we can continue to look at one or the other, if you have further questions!

Edit: I can't type.

Fascinating. Though, I would have expected that process of crowning and ratification to have been, well, like a peace treaty. How was all that accomplished without first obtaining an end to the fighting? Or how did the fighting then resume afterwards?