I don’t think so.
It is my understanding that a Consulate is an office in a foreign country established to promote (lobby for?) trade agreements and contracts, issue visas, assist in disputes (civil or legal) between it’s citizens and the host country, and stuff like that.
The official Embassy can do all that, as well, but the official Embassy is more. It is also the official residence of the Ambassador, for example. I assume the Consulate/Consul answers to the Embassy/Ambassador.
In short, a Consulate is, at most, an annex to the Embassy, but not an Embassy itself.
I presume that Consulates are established in countries where the workload is larger, or geographical distances make it more convenient to have these sub-offices.
For example, Mexico has an Embassy in Washington DC. It also has a Consulate in San Diego (or is it L.A.?), to be closer to it’s citizens in the Southern California area that need those services.
I also presume that the “rights” these consulates hold, in terms of legal/criminal jurisdictions, warrants, and such, are actually worked out in special agreements made between the two countries.
So, a Consulate will hold as much, or as little, “rights”, as the host country is willing to give it. Embassy’s seem to have a little more “juice” to them, as they have become more of an international ideal and standard.
That’s My Humble Opinion, subject to all the frailities typically found in the Human animal…