How do people with asperger's cope with a romantic relationship? Do they get romantic at all?

Depends on Severity. People with asperger's often become involved with romantic relationships, but this depends on how impaired they are in the area of social skills and relationships. Anyone who gets involved with a person with asperger's needs to be aware of their tendency towards quirky reactions along with a diminished awareness or regard for social cues.
Lack social skills . Aspergers is under pdd spectrum and the most fascinating area for researchers , their Dopamine which is the chemical for physical emotional mental balance is dysregulated great language and selective intelligence but lack social reciprocity so have a hard time with relationships but if trained their brains are remarkable , can learn easily.