AN ACR CREW OPERATES IN NON COMPLIANCE WITH DUTY TIME REGS DURING AN INTL FLT THAT INCLUDED IRREGULARITIES EXTENDING THE PLANNED OP AND AN EXTRA ENRTE STOP.

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THE ORIGINAL FLT PLAN WAS FROM ZZZ TO MDPP AND RETURN; WITH A SHOW TIME OF XA00; RETURNING AT XP20. SCHEDULED FLT TIME WAS 10 HRS 20 MINS; SO WE HAD 3 FLT CREW MEMBERS. MDPP IS AN INTL DEST AND SO DUTY LIMITS WERE THE LAST THING ON OUR MIND (VERY FEW LIMITS ON SUPPLEMENTAL DUTY TIMES RELATING TO INTL FLTS; ONLY FLT TIME LIMITS SEEM TO BE A CONCERN). DURING TKOF LEAVING MDPP; WE HAD AN ENG INDICATION ABNORMALITY; SO WE RETURNED TO THE GATE TO SEEK MAINT SUPPORT. IT TOOK US 3 HRS TO REMEDY THE PROB AND MEL A SWITCH. BEFORE THIS OCCURRED; WE HAD BEEN INFORMED OF A FUEL SHORTAGE IN THE DOMINICAN REPUBLIC; AND INFORMED US WE WOULD ONLY BE GIVEN A FRACTION OF THE FUEL REQUIRED TO RETURN TO ZZZ. THE COMPANY DECIDED THEY WANTED US TO STOP FOR FUEL AND CUSTOMS IN MIAMI. SO WITH THE AIRPLANE SIGNED OFF WE FLEW TO MIA AND LANDED. THE PAX HAD TO DISEMBARK AND CLR CUSTOMS AND REBOARD. THIS PROCESS TOOK 2+ HRS. WE CONTINUED TO ZZZ AND ENDED UP WITH APPROX 11 HRS 30 MINS FLT TIME; INCLUDING THE RETURN TO GATE FOR MAINT. THE FLT TIME WAS UNDER THE MAX; BUT DAYS LATER; LOOKING AT A FLT PRINTOUT; OUR DUTY DAY WAS 19 HRS 15 MINS (WHICH; IF STRICTLY INTL; IS FINE). I BELIEVE THAT LAST LEG FROM MIA TO DEN WAS CONSIDERED SUPPLEMENTAL-DOMESTIC; AND IF SO; WE COULDN'T EXCEED 18 HRS DUTY TIME. I THINK WE SHOULD HAVE REFUSED THE FLT. THE FLT DISPATCHERS SHOULD HAVE THOUGHT OF ALL THIS BEFORE WE STOPPED IN MIA; OR AT LEAST BEFORE WE LEFT. DISPATCH TRIED TO DO SOMETHING ALMOST IDENTICAL A FEW DAYS LATER AND WHEN THE CREW REFUSED IT; I FINALLY REALIZED WE HAD PROBABLY EXCEEDED OUR DUTY LIMITS.