It does exist, please take a look at the [amenity=embassy][1] documentation. It mentions an *diplomatic* tag to specify *whether this is an embassy, an ambassador’s residence, a consulate etc. See current values in [taginfo][2].* [taginfo][2].*
In your case please use *amenity=embassy* with the additional tag *diplomatic=consulate* is required.
*diplomatic=consulate*.
[1]: https://wiki.openstreetmap.org/wiki/Tag:amenity%3Dembassy
[2]: http://taginfo.openstreetmap.org/keys/diplomatic#values