What kind of evidence has recently made it necessary to assign the prokaryotes to either of two different domains, rather than assigning all prokaryotes to the same kingdom?

Molecular evidence

Explanation:

Carl Woese was the scientist who classified life in three domains which are bacteria, archaea, and eukarya. He used molecular evidence to classify prokaryotes into two different domains.

Carl Woese and his teammates compared the ribosomal sequence of several archebacteria with other prokaryotes and eukaryotes and found that the ribosomal sequence of archaea is more similar to eukaryotes than bacteria.

So he divided prokaryotes into two domains archaea and bacteria on this molecular evidence. So the right answer is molecular evidence.