The following variant uses the "return the true value" property of || to set a variable if it is currently false. It also uses next with a modifier to reduce nesting by a level.

The bug fix is to let the while loop run one more itteration after finding the end of the file.

use strict;

my $date;
my $lastdate;

while (($date = <DATA>) or $lastdate){
   $lastdate ||= $date;
   next if $lastdate eq $date;
   # print $lastdate report here
   print "Line $. ends date $lastdate";
   $lastdate = $date;
}
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__DATA__
10
10
10
11
11
11
12
12
12

Line 4 ends date 10
Line 7 ends date 11
Line 9 ends date 12
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Update: add bug fix explanation