• Failed with data like [ 7, 5, 3, 3] (see ++roboticus' response) and [ 8, 4, 4, 7, 6, 3 ] (see ++hdb's explanation).
• Contained some fairly crappy code (see ++BrowserUk's questions).
• Didn't test edge/corner cases such as empty array, single element array, arrays containing only zeros.

Here's the original code I posted:

Here's my take on a solution. I added some additional test data (one copied from choroba's solution).

#!/usr/bin/env perl -l

use strict;
use warnings;

use List::Util qw{sum};

my @test_arrays = ( 
    [1, 3, 8, 4], 
    [1, 6, 2], 
    [1, 3, 5, 7], 
    [4, 3, 2, 2, 1],
    [4, 3, 2, 2, 2, 2, 1],
    [5, 5, 4, 6, 2, 8, 1, 9],
);  

check_arrays($_) for @test_arrays;

sub check_arrays {
    my @full_array = @{shift()};
    print "Checking: (", join(", " => @full_array), ")";
    my $sum = sum @full_array;

    if ($sum % 2) {
        print "\tSubsets not equal";
        return;
    }   

    my $half = $sum / 2;
    my (@a1, @a2);

    for (sort { $b <=> $a } @full_array) {
        push @{(sum(@a1) || 0) + $_ <= $half ? \@a1 : \@a2} => $_;
    }

    print "\tSubsets: (", join(", " => sort { $a <=> $b } @a1),
            ") and (", join(", " => sort { $a <=> $b } @a2), ")";
    print "\tSubset sum = $half";
}
[download]

Output:

$ pm_split_equal_sums.pl
Checking: (1, 3, 8, 4)
    Subsets: (8) and (1, 3, 4)
    Subset sum = 8
Checking: (1, 6, 2)
    Subsets not equal
Checking: (1, 3, 5, 7)
    Subsets: (1, 7) and (3, 5)
    Subset sum = 8
Checking: (4, 3, 2, 2, 1)
    Subsets: (2, 4) and (1, 2, 3)
    Subset sum = 6
Checking: (4, 3, 2, 2, 2, 2, 1)
    Subsets: (1, 3, 4) and (2, 2, 2, 2)
    Subset sum = 8
Checking: (5, 5, 4, 6, 2, 8, 1, 9)
    Subsets: (1, 2, 8, 9) and (4, 5, 5, 6)
    Subset sum = 20
[download]

Here's the rewrite:

#!/usr/bin/env perl -l

use strict;
use warnings;

use List::Util qw{sum};
use Test::More;

my @test_equal_subsets = (
    [1, 3, 8, 4],
    [1, 3, 5, 7],
    [4, 3, 2, 2, 1],
    [4, 3, 2, 2, 2, 2, 1],
    [5, 5, 4, 6, 2, 8, 1, 9],
    [1, 1],
    [2, 2],
    [],
    [0],
    [0, 0],
    [0, 0, 0],
    [0, 0, 0, 0],
);

my @test_unequal_subsets = (
    [1, 6, 2],
    [7, 5, 3, 3],
    [8, 4, 4, 7, 6, 3],
    [0, 1],
    [1, 2],
    [1],
    [2],
);

plan tests => scalar @test_equal_subsets + scalar @test_unequal_subset
+s;

for (@test_equal_subsets) {
    is(check_arrays($_), 1, 'Expecting equal subsets.');
}

for (@test_unequal_subsets) {
    is(check_arrays($_), 0, 'Not expecting equal subsets.');
}

sub check_arrays {
    my $full_array = shift;

    print "Checking: (", join(", " => @$full_array), ")";

    if (! grep { $_ } @$full_array) {
        print "\tSubsets: (", join(", " => @$full_array), ') and ()';
        print "\tSubset sum = 0";
        return 1;
    }

    my $full_sum = sum @$full_array;

    if ($full_sum % 2) { 
        print "\tSubsets not equal.";
        return 0;
    }

    my $half_sum = $full_sum / 2;
    my (@a1, @a2);
    my $a1_total = 0;

    for (sort { $b <=> $a } @$full_array) {
        if ($a1_total + $_ <= $half_sum) {
            push @a1, $_;
            $a1_total += $_;
        }
        else {
            push @a2, $_;
        }
    }

    if ($a1_total == $half_sum) {
        print "\tSubsets: (", join(", " => sort { $a <=> $b } @a1),
                ") and (", join(", " => sort { $a <=> $b } @a2), ")";
        print "\tSubset sum = $half_sum";
        return 1;
    }
    else {
        print "\tSubsets not equal.";
        return 0
    }
}
[download]

Here's prove output:

$ prove pm_split_equal_sums.pl
pm_split_equal_sums.pl .. ok     
All tests successful.
Files=1, Tests=19,  0 wallclock secs ( 0.03 usr  0.01 sys +  0.02 cusr
+  0.00 csys =  0.06 CPU)
Result: PASS
[download]

Here's the full output:

$ pm_split_equal_sums.pl
1..19
Checking: (1, 3, 8, 4)
    Subsets: (8) and (1, 3, 4)
    Subset sum = 8
ok 1 - Expecting equal subsets.
Checking: (1, 3, 5, 7)
    Subsets: (1, 7) and (3, 5)
    Subset sum = 8
ok 2 - Expecting equal subsets.
Checking: (4, 3, 2, 2, 1)
    Subsets: (2, 4) and (1, 2, 3)
    Subset sum = 6
ok 3 - Expecting equal subsets.
Checking: (4, 3, 2, 2, 2, 2, 1)
    Subsets: (1, 3, 4) and (2, 2, 2, 2)
    Subset sum = 8
ok 4 - Expecting equal subsets.
Checking: (5, 5, 4, 6, 2, 8, 1, 9)
    Subsets: (1, 2, 8, 9) and (4, 5, 5, 6)
    Subset sum = 20
ok 5 - Expecting equal subsets.
Checking: (1, 1)
    Subsets: (1) and (1)
    Subset sum = 1
ok 6 - Expecting equal subsets.
Checking: (2, 2)
    Subsets: (2) and (2)
    Subset sum = 2
ok 7 - Expecting equal subsets.
Checking: ()
    Subsets: () and ()
    Subset sum = 0
ok 8 - Expecting equal subsets.
Checking: (0)
    Subsets: (0) and ()
    Subset sum = 0
ok 9 - Expecting equal subsets.
Checking: (0, 0)
    Subsets: (0, 0) and ()
    Subset sum = 0
ok 10 - Expecting equal subsets.
Checking: (0, 0, 0)
    Subsets: (0, 0, 0) and ()
    Subset sum = 0
ok 11 - Expecting equal subsets.
Checking: (0, 0, 0, 0)
    Subsets: (0, 0, 0, 0) and ()
    Subset sum = 0
ok 12 - Expecting equal subsets.
Checking: (1, 6, 2)
    Subsets not equal.
ok 13 - Not expecting equal subsets.
Checking: (7, 5, 3, 3)
    Subsets not equal.
ok 14 - Not expecting equal subsets.
Checking: (8, 4, 4, 7, 6, 3)
    Subsets not equal.
ok 15 - Not expecting equal subsets.
Checking: (0, 1)
    Subsets not equal.
ok 16 - Not expecting equal subsets.
Checking: (1, 2)
    Subsets not equal.
ok 17 - Not expecting equal subsets.
Checking: (1)
    Subsets not equal.
ok 18 - Not expecting equal subsets.
Checking: (2)
    Subsets not equal.
ok 19 - Not expecting equal subsets.
[download]