Yes. They will reference your passport if you don't have a cedula.
Also- many good clinics offer their own policies which are only good for that clinic.
The best place to ask is a reputable insurance brokerage which deals with all or most of the carriers / clinics as they will know the good and poor points about most of them.
Health insurance here in the DR is very reasonable, even in the really good clinics.
And after the past 8 months' events- beware of buying any insurance from a company that also owns a damn bank. As references for the worst, look at BanInter & Intercontinental Insurance, and BanCredito & Segna.