The quick answer is that you can't do this. If you absolutely must do something like this, then use a private setter on the class and bind things together after they are constructed (i.e. use enforced immutability instead of final fields). Hopefully it's obvious that I don't think this is a good idea - I just wanted to provide a answer to your actual question before I answer the way that I really want to.
OK - now that is out of the way, here's the real response that is called for here:
Generally speaking, this sort of situation is a strong indicator that refactoring is needed to separate concerns. In other words, the Base class is probably trying to be responsible for too many things.
I realize that the example is contrived, but think about what functionality requires the circular dependency, then factor that functionality out into a separate class/object that then gets passed to both of the Base constructors.
In complex architectures, circular dependency chains can get pretty big, but strictly forcing final fields is great way to look for those types of refactoring opportunities.
If you have a concrete example, I'd be happy to help with some refactoring suggestions to break a dependency like this.
concrete example provided - here's my suggestion:
It seems like there is a concern of obtaining an appropriate ParseStrategy based on a token. A ParseStrategyProvider. So there would be a TopLevelParseStrategy that reads the next token, looks up the appropriate parse strategy, and executes it.
TopLevelParseStrategy would hold a final reference to the ParseStrategyProvider.
The ParseStrategyProvider would then need to have a registration method (i.e. registerStrategy(token, parseStrategy) ).
This isn't functionally much different from doing this with enforced immutability via a private setter (the registerStrategy method is for all intents and purposes the same as the private setter), but the design is much more extensible.
So you'd have:
public ParseStrategy createParser(){
ParseStrategyProvider provider = ParseStrategyProvider.create();
TopLevelParseStrategy topLevel = new TopLevelParseStrategy(provider);
provider.registerStrategy("(", topLevel);
// create and register all of your other parse strategies
return topLevel;
}