open(my $fh, \'>\', $path) || die $!;
my_sub($fh);
Can my_sub() somehow extrapolate $path from $fh?
You can call stat or IO::Handle::stat on a filehandle -- that will give you the device and inode of the file that you have opened. With that and a little operating system wizardry you can find the filename. OK, maybe a lot of operating system wizardry.
The find command has an -inum option to find a file with a specified inode number. This is probably not going to be as efficient as caching the path when you open the file, as gbacon recommends.