Redirecting stdout+stderr such that both get written to a file while still outputting to stdout is simple enough:
cmd 2>&1 | tee output_file
>
{ { cmd | tee out >&3; } 2>&1 | tee err >&2; } 3>&1
Or, to be pedantic:
{ { cmd 3>&- | tee out >&3 2> /dev/null; } 2>&1 | tee err >&2 3>&- 2> /dev/null; } 3>&1
Note that it's futile to try and preserve order. It is basically impossible. The only solution would be to modify "cmd" or use some LD_PRELOAD or gdb hack,