How do I retrieve the values from the hash now?

Use the same expressions you used for assignment.

$TGS_Value{$tgs}{VAR} = \@VAR;
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After that, $TGS_Value{$tgs}{VAR} is an array reference.

my $var_ref = $TGS_Value{$tgs}{VAR};
my @VAR = @{$var_ref};
# ...is the same as:
my @VAR = @{ $TGS_Value{$tgs}{VAR} };
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