In any context, anti- means against or opposite. In this case, it would be the opposite of or that which is against a romantic comedy. If we know the elements of the romantic comedy genre, we can determine the features of its antithesis, the anti-romantic comedy.
A romantic comedy is a love story, generally centered around a series of humorous circumstances and characters, where love is thwarted through external events and misunderstandigns but which finally ends happily. There are plenty of them around in every form, ranging from Shakespeare's romantic comedies to more modern romantic comedy movies such as You've Got Mail; While You Were Sleeping; My Big, Fat Greek Wedding; and Notting Hill.
An anti-romantic comedy still has to be humorous and it must still have an element of romance; if it did not, it would not be categorized as it is. So, that leaves two elements which change in an anti-romantic comedy--how love is thwarted and the happy ending. Though it is possible to have a comedic romantic journey without a happy ending, it is not done often and the elements comprising how love is thwarted are of a specific nature.
In an anti-romantic comedy, the boy does not get the girl--or vice versa--but the thwarted love comes from within the character's own tormented psyches instead of from external influences, misunderstandings and complications as in romantic comedy. The parting of their ways must seem amicable or inevitable, but a reflection of reality, in order to satisfy an audience which is expecting a comedy of sorts. Modern Romance is a filmatic example of an anti-romantic comedy: the hero is kept from love by his own psychological "issues" and realism.