These schemes achieve a security definition called OAE2 (STREAM specifically achieves nOAE, which Rogaway proves equivalent to OAE2) and are robust against reordering and truncation attacks. For more information, please see the paper:
Ah, thanks for the reply Tony. This would indeed prevent the problem I described. Kinda curious about the downvotes, since tptacek's original comment suggested nothing like CHAIN or STREAM, but on crypto HN you gotta roll with the punches.