Okay, I think CheeseLord has it cracked. As he suggests, closing the STDIN filehandle and reopening it on the /dev/tty (or /dev/pts/xx or whatever) device from which the script was called works a treat. Thanks very much!

(Update): i.e.

use POSIX;
close(STDIN);
my $tty=POSIX::ctermid();
open(STDIN,"$tty");
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Thanks very much!