Malaysia has secular and religious courts. They also have secular and religious police. You only have to worry about the “sin police” if you are a Muslim.
Apart from Saudi and Iran, AFAIK just about all Muslim countries have secular laws on the books and secular law courts parallel to the shar‘î courts. Tunisia and Turkey, IIRC, are anomalous in not having any Islamic courts. The usual modus vivendi in most Muslim countries is to apply Islamic law in family-related matters but secular law for most other things in general, like traffic courts, criminal courts, etc. I’m not sure about probate courts. This is a big generalization, and no doubt varies in the details from country to country, but Islamic law application in most countries has been restricted to a relatively narrow range.
Well sure, we have “Secular” laws (like traffic rules) here, but no “Secular” courts. The only consitution here is Sharia, the testimony of Jews and Christians (and women) are discounted and so on.
So in a word, no there are no courts here you would consider non-religious.
As a cultural thing however, the Saudis try to avoid the courts. The general iea is to settle darn near anything between themselves.
Jomo Mojo has it pretty well. The extent of penetration of civil/religious courts and their spheres of action seem to depend on whether the particular government has adopted all or part of shar’ia as their official legal code, on how much do they want/care for the mullahs vs. the State to be in control of the system, and on how centralized IS control of the State.
You also have federal nations like Nigeria and Malaysia, where individual states may adopt shar’ia, so if you sought “secular” justice you’d have to go before a Federal Court… IF there is federal jurisdiction for your case (If not, you get the stones heaped upon ya).
What also happens often even under nominally secular governments is that, if it’s not a crime against the State, everyday justice/dispute resolution may be handled by “council-of-village-elders”-type methods.
They plan here to set up an international free zone for financial companies, with its own regulatory framework based on the best of “western” commerical legal systems.
However I am not convinced that that could protect an overseas company in a dispute with a local company - even if the local company was supposed to abide by the free zone laws, what is to stop them having a local sharia court make its own ruling?