We don’t live in a world where that is a realistic proposition. The EU not federalising means it cannot ever be a pole in a multipolar world in the way that America and China are.
I suspect that there would be something between the two to start with.
First of an agreement to fund a joint control command system to facilitate a more unified response when military action is required (similar to some of the systems NATO has, just not reliant on the US).
Secondly an agreement for such a system to be able to draw on the resources required as needed.
The second part may find resistance from certain sections, however there's likely to be enough support (although there may be limits in place, such on how much can be called on without the nation in question agreeing further - for example if a force over a certain size is to be deployed then it must be drawn from more than a set number of nations without consent from the nation being asked).
For example, a third party country starts attacking an EU state or EU aligned state (the latter could include the UK or even Canada) and a force of 50,000 was to be sent to aid them. The rules may state that no one nation should be expected to provide more than 5% of their personal without further agreement. However it may well be that France agrees that due to the attack being on a country fairly close to them (unlikely, but that's the example I'm going with) that they will provide 7% of their personal so the depotment can happen faster.