#!/usr/local/bin/perl -w
use strict;

LINE: while ( <DATA> ) {
  $_ .= <DATA> if /<B/;
  next LINE if (m/\.{4}<B.*ird>/s .. /(\+\=\+)$/);
  print;
}
__END__

(Example 1)
This is Example 1
This is some text....<Bluebird>..
MBAAAEgAAAQAB
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+

(Example 2)
but sometimes it looks like this:
This is some text....<B
luebird>..MBAAAEgAAAQAoBA
AAQKAREDSCETRTBDFS
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+