Splitting from #4711 since there are two sets of work here - one to support in the base adapters, and one to update the Flutter adapter.
Here's an example of the issue in Dart:
import 'dart:io';
void main() {
print('stdout');
stdout.write(StackTrace.current);
print('stderr');
stderr.write(StackTrace.current);
}

In the old debug adapters it looks like:
