It's a trick question, reality and conspiracy theory are orthogonal. Some conspiracy theories are true and are backed by evidence. Some conspiracy theories are true but nobody can find the evidence. Some conspiracy theories are false and have no evidence. Some conspiracy theories are false but there is nevertheless some evidence that seems to support them. Explanatory examples:

True conspiracy theory backed by evidence: Criminal prosecutors present a theory about a gang of bank robbers conspiring to rob a bank. They have a lot of phone records and surveillance footage as evidence.

True conspiracy theories with no evidence: Crimes with two or more participants which have successfully evaded discovery. Somebody smells a rat but has nothing but their gut to go off of. Distinguishing these from false conspiracy theories is impossible (so no example), but that doesn't mean it didn't happen. Sometimes people get away with it.

False conspiracy theory with no evidence: wolverine876 is in league with Martians to take over Earth.

False conspiracy theory with some evidence: To understand how this could happen, you must understand that evidence is not the same as proof. Circumstantial evidence and eye witness testimony can often point in the direction of something that isn't actually true. In the case of the Second Gulf of Tonkin incident, radar and sonar evidence lead the Navy to believe that it was under attack from the North Vietnamese. Turns out they were almost certainly shooting at shadows. They believed something that the evidence then available to them supported, but it turned out to be wrong.