Those who believe that the Trojan War was real point to the archaeological evidence of the city of Troy, which was discovered by Heinrich Schliemann in the 19th century. They also cite the fact that the war is mentioned in several Greek texts, including Homer's Iliad and Odyssey.
Those who believe that the Trojan War was not real argue that the archaeological evidence is inconclusive and that the Greek texts are not reliable sources of history. They also point to the fact that there is no mention of the Trojan War in any non-Greek sources.
Ultimately, the question of whether or not the Trojan War was real is a matter of opinion. There is no definitive evidence to support either side of the argument.