This turned out to be the same issue as this one, in a slightly different guise.
In short, I wasn't keeping a reference to the NSWindow object, but instead was letting it go out of scope. So the GUI window would stick around for a while, but eventually (after some number of events or other code creating behind-the-scenes garbage) it is noticed and disposed of by the garbage collector. The window is then torn down.
It's all perfectly reasonable once you think of it, and happens under both Xamarin and MonoMac (just at slightly different times).
The simple solution, of course, is to retain a reference to the window until you're truly done with it. Problem solved!
(And yes, I feel a bit sheepish, but hopefully this question will get found by future Mac C# developers, and save them some grief.)