Precedence issue, so use parens to disambiguate.

grep /testme/, @array ? qq! yes ! : qq! no !
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means

grep /testme/, ( @array ? qq! yes ! : qq! no ! )
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You want

grep(/testme/, @array) ? qq! yes ! : qq! no !
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or

(grep /testme/, @array) ? qq! yes ! : qq! no !
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Update: AnomalousMonk mentioned "Perhaps make clear that both solutions need to be enclosed in original set of parens to work."

Replacing

grep /testme/, @array ? qq! yes ! : qq! no !
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with

(grep /testme/, @array) ? qq! yes ! : qq! no !
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means

my $string = qq! first !
        . ( grep /testme/, @array ? qq! yes ! : qq! no ! )
        . qq! end !;
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becomes

my $string = qq! first !
        . ( (grep /testme/, @array) ? qq! yes ! : qq! no ! )
        . qq! end !;
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This may be another case of missing the forest for the trees, but what is going on in the version in lines 27-30?

#!/usr/bin/perl
use warnings;
use strict;
# [id://863649] 

print "\n\t First test: incorporating correction by ikegami\n";

my @array = qw( testme foo bar );
my $i = 1;

for my $string(@array) {
    my     $test = $string;
    my $string2 = qq! first ! . (grep /$test/, $string) ? qq! yes ! : 
+qq! no !    . qq! end !;
    print "\$test: $test | \$string: $string | \$string2: $string2 | $
+i \n";
    $i++;
}
print "\n\t now testing with 'blivitz'\n";
for my $string(@array) {
    my     $test = "blivitz";
    my $string3 = (grep /$test/, $string) ? qq! yes ! : qq! no !;
    print "\$test: $test | \$string3: $string3 | $i \n";
    $i++;
}

print "\n\t BUT...\n";

my $string = qq! first !
        . (grep /blivitz/, @array) ? qq! yes ! : qq! no !
        . qq! end !;
print $string . " -last test, lines 27-30 \n";

=head OUTPUT of 863649.pl as corrected by [ikegami] and extended


        First test: incorporating correction by ikegami
$test: testme | $string: testme | $string2:  yes  | 1
$test: foo | $string: foo | $string2:  yes  | 2
$test: bar | $string: bar | $string2:  yes  | 3

         now testing with 'blivitz'
$test: blivitz | $string3:  no  | 4
$test: blivitz | $string3:  no  | 5
$test: blivitz | $string3:  no  | 6

         BUT...
 yes  -last test, lines 27-30

=cut
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WTF?!? grep is finding "blviitz" in @array? (or, ww is having another senior moment?)

Note the lack of 'first' and 'end' in the output.

my $string = qq! first !
        . (grep /blivitz/, @array) ? qq! yes ! : qq! no !
        . qq! end !;
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is

my $string = 
   (  qq! first ! . (grep /blivitz/, @array)  )
      ? ( qq! yes ! )
      : ( qq! no ! . qq! end ! );
[download]

You removed a set of parens. See the update I made at the same time you were posting.