I have to pass the file (with the two strings) to other subroutines

Do you mean the file handle? For example:

use warnings;
use strict;
use File::Temp;

sub mysub {
    my ($fh, $str1, $str2) = @_;
    $fh->seek( 0, 0 );    
    print <$fh>;

}


my $fh = File::Temp->new();
my $s1 = 'This is string one';
print $fh "$s1\n";
my $s2 = 'This is string two';
print $fh "$s2\n";

mysub ($fh, $s1, $s2);
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Gives:

This is string one
This is string two
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