Is it possible to derive a new 12 word seed from a 12 word seed with passphrase (13th word) such that they both produce the same master key?
No, this is not possible.
The mnemonic and passphrase are put through PBKDF2. Any change in the input results in a completely different output.
While it is possible that there exists some 12 word mnemonic that just happens to hash to the same seed as your original mnemonic + passphrase, this would require bruteforcing all possible mnemonics to discover, and would not be derivable from the original mnemonic in any form - a hash collision would be completely unrelated.
