#!/bin/bash

for i in {1..5}; do
  printf '%s:%s\n' "$((++c))" "out $i"
  printf '%s:%s\n' "$((++c))" "err $i" >&2
done

$ ./s
1:out 1
2:err 1
3:out 2
4:err 2
5:out 3
6:err 3
7:out 4
8:err 4
9:out 5
10:err 5

$ ./s | sort -n
2:err 1
4:err 2
6:err 3
8:err 4
10:err 5
1:out 1
3:out 2
5:out 3
7:out 4
9:out 5

$ ./s > output_all 2>&1
$ sort -n output_all
1:out 1
2:err 1
3:out 2
4:err 2
5:out 3
6:err 3
7:out 4
8:err 4
9:out 5
10:err 5

$ fgrep :err output_all | sort -n
2:err 1
4:err 2
6:err 3
8:err 4
10:err 5

% cat s
#!/usr/bin/bash

for i in {1..5}; do
  printf '%s\n' "out $i"
  printf '%s\n' "err $i" >&2
done

% cat unbuffer 
#!/bin/sh
#
# Description: unbuffer stdout/stderr of a program
# Author: Don Libes, NIST
#
# Option: -p  allow run program to read from stdin (for simplification)
#
# Note that expect can 'continue' a comment line, so the follow re-runs this
# as a expect command regardless of its PATH location.
# \
exec expect -- "$0" ${1+"$@"}

if {[string compare [lindex $argv 0] "-p"] == 0} {
   # pipeline
   set stty_init "-echo"
   eval spawn -noecho [lrange $argv 1 end]
   interact
} else {
   set stty_init "-opost"
   eval spawn -noecho $argv
   set timeout -1
   expect
}

% unbuffer ./s
out 1
err 1
out 2
err 2
out 3
err 3
out 4
err 4
out 5
err 5
% unbuffer ./s | fgrep err
err 1
err 2
err 3
err 4
err 5
% unbuffer ./s | fgrep out
out 1
out 2
out 3
out 4
out 5