Proving similarity between two rankings

We are seeking a comparison of two different rankings of the same 3 systems

Background: We have conducted a test of 3 different systems. Participants experienced a system and rated it on a 7-point likert scale, this was repeated for each system. At the end of the test they were tasked with ranking the systems relative to each other from best to worst, during this they were not able to see their previous likert ratings. The likert ranking has then been converted to a standard competition ranking, making both rankings range from 1 to 3.

Our question is then which test we can use to see whether the rankings match or not. Ideally we would like to be able to say that the two rankings are significantly similar.

Since the likert ranking has been converted (we still have the original if we need it to be converted otherwise) some of the entries contain the same numbers, due to equal likert scores. E.g. if the likert scores were 5, 5, 3 it has then been converted to 1, 1, 3.