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Today, I read that Turkish president Erdogan wants the Saudi hit squad that killed journalist Jamal Khashoggi to face justice in Turkey for their role in the murder.

There seems little doubt that a murder occurred; there seems little doubt that the murder occurred inside the Saudi consulate in Turkey. If that is the case, then the murder occurred on Saudi soil, and thus the perpetrators should be facing Saudi justice - not Turkish.

I'm not at all interested in the politics of the case, only the legal jurisdiction of a case like this. Since the case is ongoing and facts may change, for purposes of this question, assume that it is the case that a murder occurred in the Saudi consulate, and the perpetrators were Saudi nationals.

Is it true that consuls are territories of the countries they represent, and not of the area they take up? Where do crimes get adjudicated in cases like this? Are there different protocols to handle certain kinds of crimes? Does it depend on a treaty between two countries, or is this standard international law?

According to Wikipedia's article Diplomatic Mission:

Contrary to popular belief, most diplomatic missions do not enjoy full extraterritorial status and – in those cases – are not sovereign territory of the represented state. Rather, the premises of diplomatic missions usually remain under the jurisdiction of the host state while being afforded special privileges (such as immunity from most local laws) by the Vienna Convention on Diplomatic Relations. Diplomats themselves still retain full diplomatic immunity, and (as an adherent to the Vienna Convention) the host country may not enter the premises of the mission without permission of the represented country, even to put out a fire.

(Supporting citations omitted)

According to its article on Vienna Convention on Diplomatic Relations key provisions include:

Article 22. The premises of a diplomatic mission, such as an embassy, are inviolable and must not be entered by the host country except by permission of the head of the mission. Furthermore, the host country must protect the mission from intrusion or damage. The host country must never search the premises, nor seize its documents or property.

Article 29. Diplomats must not be liable to any form of arrest or detention. They are immune from civil or criminal prosecution, though the sending country may waive this right under Article 32.

However, people who are not diplomats do not have automatic protection from arrest, and the idea that an embasy is the "soil" of the country it represents is largely obsolete. See This stack exchange politics question.

So if Turkey were to obtain custody of people it accused of doing the killing, and if those people were not accredited diplomats, it could try them under Turkish law. Whether any of that is likely to happen is another question.

The answer of @David Siegel is correct as far as it goes.

I would further venture the opinion that it is very likely that even though the Saudi Arabian embassy is not the territory of Saudi Arabia, that diplomatic immunity would very likely pose a bar to the prosecution of at least some of the defendants in a case if one was brought in Turkish courts and would make prosecution of a criminal case in the Turkish courts as a practical matter very difficult. It is possible that some of the defendants, however, would lack diplomatic immunity.

His answer does not address the further question of whether Saudi Arabia would have jurisdiction to try and criminally punish Saudi Arabian officials who committed a murder at a Saudi Arabian diplomatic complex in Turkey. The answer is that it would have jurisdiction to do so.

But, there are strong indications that the murder was committed by Saudi Arabian officials under official lawful orders from the superiors of those officials with the authority to give those orders (e.g. someone in the Crown Prince's office with whom there were at least four phone communications with the presumed murderers that day):

The Saudi entourage who went to the embassy in Turkey to cut off
journalist Jamal Khashoggi's fingers, inject him with a drug to
silence him, and dismember him with a bone saw made four calls that
day to Saudi Crown Prince Mohammed bin Salman's office, according to
Turkish media reports. The Crown Prince denies knowing anything about
the gruesome torture/murder of Khashoggi, who was a US resident. Three
of Khashoggi’s children are US citizens.

Those official orders provide provide a valid legal defense under Saudi Arabian law (to the extent that Saudi Arabia as an absolute monarchy can even be said to have rule of law in a complete and meaningful sense) to charges under Saudi Arabian jurisdiction that those officials engaged in murder.

Outside the criminal justice process, there are a variety of diplomatic and military options available to the Turkish government.

The most obvious is that it could (and likely will) expel the diplomats involved from Saudi Arabia and possibly the entire diplomatic mission from Saudi Arabia from Turkey. It could also withdraw its own diplomats from Saudi Arabia.

Turkey would also very likely be considered justified in its actions by the community of nations if it authorized the used of military force including summary assassination against the Saudi Arabian individuals it finds to have been involved, outside the criminal justice process as a political determination, although this justification would likely not extent to the Crown Prince of Saudi Arabia even if Turkey founds that he gave an order lawful under Saudi Arabian law directing the torture and murder of this individual.

The Turkish government could also probably have a basis for pushing a war crime type prosecution of at least some of the people involved.

More extremely, Turkey could declare war on Saudi Arabia generally in retaliation for this action, although this level of escalation would be considered disproportionate by most international observers standing alone.

Also, because the victim was a lawful U.S. resident, employee of a U.S. newspaper, and parent of U.S. children, it isn't inconceivable that the United States government, as well as the Turkish government, would have standing to take diplomatic or legal action against the Saudi Arabian government, although again, diplomatic immunity might bar a criminal prosecution against some of the defendants. (The issue of whether diplomatic immunity protects diplomats from prosecutions under the laws of countries other than the host country is to the best of my knowledge not a question governed by clear precedent.)