What cryptanalysis is possible against two independent keystreams XORed against plaintext?
Asmuth and Blakley provided a proof that, assuming the keys for each cryptosystem are chosen independently, breaking their composite cryptosystem is at least as hard as breaking the hardest part of either. [1]  Building on their work, cascade ciphers have been shown to in fact be harder to break than the hardest part of either.

Admittedly, what you're talking about sits in an awkward place between cryptographic exponentials and cascade ciphers.  A cryptographic exponential (of two ciphers) is defined as:  $$M\oplus R\oplus R'\&gt;\|\&gt;E(R)\&gt;\|\&gt;E'(R')$$ where R and R' are truly random bitstrings.

I can't give much of anything concrete other than that.  Hope this answers your question.


C. Asmuth, G. Blakley.  "An efficient algorithm for constructing a cryptosystem which is harder to break than two other cryptosystems"