A C525A FO; WHILE PERFORMING THE TAXI AFTER LNDG AT ILM; TAXIED WITHOUT CLRNC ONTO AN ACTIVE TXWY.

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I WAS PF. THE EVENT OCCURRED JUST AFTER LNDG AT ILM RWY 17 AND EXITING ON H TXWY. AFTER EXITING; WE WERE LOOKING DIRECTLY INTO THE SETTING SUN; AND I CONTINUED TO LET THE ACFT MOVE TOWARD THE CUSTOMS RAMP; STRAIGHT AHEAD; WHICH WAS OUR DEST. THE PNF MADE ATTEMPTS TO CONTACT GND; BUT THE FREQ WAS BUSY WITH OTHER TFC. I CONTINUED TO TAXI; NOT SURE IF I WAS PAST THE 'HOLD SHORT/CLR OF RWY' MARKINGS OR NOT. THE GND CTLR THEN ADVISED US THAT WE WERE TAXIING WITHOUT CLRNC ACROSS AN ACTIVE TXWY. LOOKING STRAIGHT AHEAD AT THE CUSTOMS RAMP; I HAD NOT SEEN A TXWY. IN ACTUALITY; TXWY A CROSSES THE E SIDE OF THE RAMP. I DO NOT REMEMBER SEEING A TXWY SIGN WHILE ON TXWY H FOR TXWY A. WE THEN GOT CLRNC TO CONTINUE TAXI. FACTORS AFFECTING THIS OCCURRENCE WERE: POOR VISIBILITY DUE TO SUNSET MISSING TXWY SIGN? CONGESTED FREQ; MY INDECISION WHETHER I WAS CLR OF ACTIVE RWY; AND UNFAMILIARITY WITH ARPT DIAGRAM. WE WERE NOT IN IMMEDIATE CONFLICT WITH ANY ACFT AND WERE NOT TOLD TO CONTACT ANYONE AFTER THE OCCURRENCE.