No problem, by now I've sort of guessed you have your brain in the right place ;)
I'm serious about my post though: not being able to swiftly respond to each other or to handle/settle things in order of appearance/chronology is a major drawback of this style of play - speaking as someone with 17 years of face-to-face RPG experience.
In this case, since our actions didn't overlap - I was looking for something useful to confiscate by showing my badge, seeing how far I could take being an agent - there's indeed no problem whatsoever. Once we have two or more players aiming for the same objective (or intertwined ones), we need to be realistic about this, or true to the setting at least.
But I know you'll all agree with me on this, so... just unnecessarily elaborating on my point, I guess. And I wanted you to know what I was planning, since MkLear and Hex would probably have hated my guts for it :))