Nah, full-on babelfish is simply not possible. The meaning of the beginning of a sentence can be modified retroactively by the end of the sentence. This means that the Babelfish must either be greatly delayed or awkwardly correct itself every once in a while.
Ah, reminds me of learning German, where you can chuck all the verbs onto the end. There was this sentence we had as a fun toy example, where it was like half a paragraphs of verbs in the end, and you had to try and match them up with the beginning of the sentence.
Joke that I heard was supposed to come from the 1880s.
Some travelers stop off at an inn in the Swiss Alps, and they notice that are the various tables at the inn are other nationalities. They amuse themselves by listening in on the conversations (and displaying their stereotypes) - the Italians are all talking at the same time very loud and never stop moving their hands, the French are all arrogant artistes, the Danes are boring and only talk about the weather, and then they notice the Germans.
There was obviously a very important conversation going on with the Germans because one German would say something and every other German would stop and listen intently until they were done, and then another one would start up and every one would stop and listen intently until that one was done, but in following the conversation it became clear it was not because the conversation was important, but because the Germans were waiting to hear the verb to understand what was being said.
I wonder if an esolang exists that could be a universal target though (if a language can handle any ambiguity by appending, then it could always be output without backtracking).
