The end result? A user starts with the password "Seekrit1" when they join the company, then the next month they change to "Seekrit2", "Seekrit3", and so on.
This is hardly what the security policy intended, so stopping that happening isn't actually a bad idea.
Obviously storing a complete password history in plaintext (which would even have to be online for the consultants' plan to work) is ridiculous, but pre-calculating a bunch of hashes of similar passwords every time a new password is set would certainly be feasible.
Changing passwords is usually a rare occurrence, but making this at least slightly computationally-expensive shouldn't be a problem. Of course, the more expensive the password-hashing algorithm, the fewer "similar" passwords you would reasonably be able to pre-calculate, which is an odd trade-off to have to make a call on.
One thing to be wary of, though: by pre-calculating and storing a bunch of hashes of similar strings, this opens you up to (what might be called) a related plaintext attack. I have no idea if existing password hashes are specifically designed to be resistant to this, but would guess not. (edit: I didn't think this sentence through correctly. Thanks, Joachim)