some way of telling the source process to send everything to the same filehandle
open(STDERR,'>&', STDOUT); creates a new filehandle (so it doesn't help), but *STDERR = *STDOUT; makes both STDOUT and STDERR refer to the same filehandle. You don't even need to turn off buffering.
use IO::Handle ();
open(STDERR,'>&', STDOUT);
print("STDOUT = ", fileno(STDOUT), "\n"); # 1
print("STDERR = ", fileno(STDERR), "\n"); # 2
print STDOUT 'a';
print STDERR 'b';
print STDOUT 'c';
STDOUT->flush(); # ac
STDERR->flush(); # b
print("\n");
*STDERR = *STDOUT;
print("STDOUT = ", fileno(STDOUT), "\n"); # 1
print("STDERR = ", fileno(STDERR), "\n"); # 1
print STDOUT 'a';
print STDERR 'b';
print STDOUT 'c';
STDOUT->flush(); # abc
STDERR->flush();
Of course, this doesn't work if you fork.
If you can't modify the program, you can replace
perl script.pl
with
perl -e "*STDERR = *STDOUT; do 'script.pl'"