LTT MAKES ILS APCH BELOW MINIMUMS.

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FIRST LEG OF TRIP DFW TO TYR. ON DEP DFW; WX AT TYR WAS RPTED CLR AND 15 MI VISIBILITY. UPON ARR WE OBSERVED A GND FOG WAS ROLLING IN AT TYR; WITH TOPS 20-40 FT AGL. WE ANTICIPATED POSSIBLE FOG AS WELL ON OUR NEXT LEG TO GGG AS ITS LOCATION WAS ONLY 35 MI AWAY. WE TOOK ON ADDITIONAL FUEL; IN CASE WE NEEDED TO HOLD. WX AT GGG WAS ALSO RPTING CLR AND 15 MI VISIBILITY WHEN WE DEPARTED TYR. UPON BEING TURNED OVER TO GGG APCH AT 14 DME OFF GGG; APCH CTL RPTED WX AT GGG 3/8 MI VISIBILITY; WINDS CALM. CAPT SAID; 'WE HAVE GGG ARPT IN SIGHT. FOR PURPOSES OF SHOOTING THE APCH; BASED ON A PIREP FROM US; WOULD YOU CONSIDER CALLING THE ARPT VISIBILITY 1/2 MI. WE HAVE TO HAVE 1/2 MI TO SHOOT THE APCH.' TWR RESPONDED; 'UH! YEAH! SURE. ARPT VISIBILITY 1/2 MI; HOWEVER; BE ADVISED LAST RPT SHOWED 3/8 VISIBILITY 2200 RVR. YOU ARE CLRED FOR THE ILS RWY 13 GGG.' (THIS RPT WAS AT 10 DME.) I TOLD THE CAPT I WAS NOT COMFORTABLE SHOOTING THE APCH. AT THIS POINT; WE ARE LOC INBOUND AT 2500 MSL. HE SAID; 'YOU'RE DOING FINE. CONTINUE THE APCH. RWY IN SIGHT.' OUTSIDE FAP AT APPROX 6 DME; TWR STATED; 'GGG RWY RVR 2200.' CAPT RESPONDED 'WE ARE DOING THIS APCH PREDICATED ON THE 1/2 MI VISIBILITY YOU GAVE US PREVIOUSLY. TWR SAID; 'ROGER; YOU'RE STILL CLRED FOR THE ILS 13.' I STATED TO THE CAPT ONCE AGAIN I WAS NOT COMFORTABLE SHOOTING THIS APCH. HE TOLD ME TO CONTINUE. 'YOU'RE DOING FINE; JUST CONTINUE ON.' NOT WANTING TO MAKE WAVES BECAUSE I'D EXPERIENCED PREVIOUS CLASHES WITH THIS CAPT IN THE PAST; AND FEELING INTIMIDATED; I CONTINUED THE APCH. I WAS CONCENTRATING ON THE APCH AND MAKING SURE ALL THE CHKLISTS WERE PERFORMED. ALL I HEARD AT THE TIME WAS '1/2 MI -- YOU'RE CLRED FOR THE ILS.' I DID NOT REALIZE AT THE TIME I MIGHT BE CONSIDERED SHOOTING AN APCH ILLEGALLY (1/2 MI OR 2400 RVR MINIMUM REQUIRED). AFTER CAREFUL ANALYSIS ONCE ON THE GND WE SHOULD HAVE BROKEN OFF THE APCH ONCE THE RVR WAS RPTED 2200. WE SHOULD HAVE HELD UNTIL THE TWR OFFERED 1/2 MI OR 24000 RVR.