I'm not sure just what you are looking for, but try something like:

c:\@Work\Perl\monks>perl -e
"use warnings;
 use strict;
 ;;
 use List::MoreUtils qw(uniq);
 use Data::Dumper;
 ;;
 my @array1 = ('ux_prim_clk', 'ux_side_clk', 'ux_xtal_frm_refclk', 'pr
+im_ibbs');
 my @array2 = ('ccu_ux_xtal_frm_refclk_ack', 'ibbs_ux_prim_clkack', 's
+br_ux_side_clkack');
 ;;
 my @array1_split_2D = map [ split /_/ ], @array1;
 my @array2_split_2D = map [ split /_/ ], @array2;
 ;;
 print Dumper \@array1_split_2D;
 print Dumper \@array2_split_2D;
 ;;
 for my $ar_array_1_cols (@array1_split_2D) {
   for my $ar_array_2_cols (@array2_split_2D) {
     my %array_2_colums = map { $_ => 1 } @$ar_array_2_cols;
     my @intersection = grep exists $array_2_colums{$_}, uniq @$ar_arr
+ay_1_cols;
     ;;
     print qq{(@$ar_array_1_cols) elements in (@$ar_array_2_cols): \n}
+;
     if (@intersection) {
       print qq{  '$_'} for @intersection;
       }
     else {
       print '  - no intersection -';
       }
     print qq{\n};
     }
   print qq{\n};
   }
"
$VAR1 = [
          [
            'ux',
            'prim',
            'clk'
          ],
          [
            'ux',
            'side',
            'clk'
          ],
          [
            'ux',
            'xtal',
            'frm',
            'refclk'
          ],
          [
            'prim',
            'ibbs'
          ]
        ];
$VAR1 = [
          [
            'ccu',
            'ux',
            'xtal',
            'frm',
            'refclk',
            'ack'
          ],
          [
            'ibbs',
            'ux',
            'prim',
            'clkack'
          ],
          [
            'sbr',
            'ux',
            'side',
            'clkack'
          ]
        ];
(ux prim clk) elements in (ccu ux xtal frm refclk ack):
  'ux'
(ux prim clk) elements in (ibbs ux prim clkack):
  'ux'  'prim'
(ux prim clk) elements in (sbr ux side clkack):
  'ux'

(ux side clk) elements in (ccu ux xtal frm refclk ack):
  'ux'
(ux side clk) elements in (ibbs ux prim clkack):
  'ux'
(ux side clk) elements in (sbr ux side clkack):
  'ux'  'side'

(ux xtal frm refclk) elements in (ccu ux xtal frm refclk ack):
  'ux'  'xtal'  'frm'  'refclk'
(ux xtal frm refclk) elements in (ibbs ux prim clkack):
  'ux'
(ux xtal frm refclk) elements in (sbr ux side clkack):
  'ux'

(prim ibbs) elements in (ccu ux xtal frm refclk ack):
  - no intersection -
(prim ibbs) elements in (ibbs ux prim clkack):
  'prim'  'ibbs'
(prim ibbs) elements in (sbr ux side clkack):
  - no intersection -
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See List::MoreUtils or List::Util for  uniq() (which may not be strictly necessary) and Data::Dumper (which is a core module) for  Dumper.

Update 1: See also How do I compute the difference of two arrays? How do I compute the intersection of two arrays? in perlfaq4 (you seem to be looking for an intersection).

Update 2: Changed example code to show extraction of intersection elements to  @intersection array separately from display of the intersection.