Store stdout and stderr in file, then recreate the output later

I've redirected output to a file like so:

$ ./test.bash 2> >(sed 's/^/stderr: /' >> output) > >(sed 's/^/stdout: /' >> output)
$ cat output
  stdout: Hello World!
  stderr: error

I would like to precisely reproduce the output of the original command ./test.bash. This is what I tried:

$ cat output | tee >(grep '^stdout: '|sed 's/^stdout: //') >(grep '^stderr: '|sed 's/^stderr: //' 1>&2) >/dev/null 2>&1 | cat
  error
  Hello World!

Obviously, I would like the output to be in the right order. What is the best way to go about doing this?