"evidentiary support" refers to the evidence or proof that backs up or supports a claim or statement. It means having facts, information, or data that can be used to prove or demonstrate something. Full definition
Using similar criteria to the What Works Clearinghouse, the RAND team created four tiers of evidentiary support. (questiontank.com)
It may be futile to try to clarify when fear lacks evidentiary support, or — as is often the case — simply exists in the absence of evidence one way or the other, but it's surely worth trying. (dotearth.blogs.nytimes.com)