Prelude: This code is crazy. If you actually have code in your app like this, fix it now. Either make them both protected or both public!
Regarding the error: The CLR has a lot of really strange 'edge case' rules in it for dealing with things like this. The best place to look for this kind of stuff is usually Eric Lippert's blog.
In saying that though, it looks like mono is actually doing the more sensible thing here in my opinion.
On second look, the C# one makes more sense once you factor in the 'behind the scenes' stuff.
Properties are not "first class" in MSIL. A property in C# or VB is just compiled down to a get and set method (the compiler also sticks an attribute somewhere for bookkeeping).
int MyField { get; set; }
will actually produce MSIL for two methods:
void set_MyField(int value);
int get_MyField();
Now, given that your new
method has a different type, you'll end up with the following 2 setter methods.
void set_MyField(int value);
void set_MyField(string value);
When you call x.MyField = "string"
you're just calling one of those methods. This then boils down to a normal method overloading scenario. It's perfectly valid to have two methods with the same name that take different parameters, so the compiler will just select the string one and carry on it's merry way.
So yeah. The C# one makes sense if you know how the internals work, the Mono one makes more sense if you don't.
Which one is "more correct"? Ask Eric Lippert :-)