How to analyze ranks as responses?

A client has data where multiple subjects (people) have been presented with with four product alternatives, and after experiencing all four, were asked to rank the products.

The data look like this:

Person  Variety Rank
Matt    A       3
Matt    B       4
Matt    C       1
Matt    D       2
Aliyah  A       3
Aliyah  B       2
Aliyah  C       1
Aliyah  D       4
Manny   A       4
Manny   B       3
Manny   C       2
Manny   D       1
Theo    A       3
Theo    B       1
Theo    C       2
Theo    D       4
...     ...     ...


How would one analyze this?

It was suggested here to do a Friedman Test, which is the non-parametric version of a repeated measures ANOVA, in this case subject would be block. The Friedman test converts responses (continuous or ordinal) to ranks within each block/subject, and then calculates a statistic to compare against the Chi-Square distribution.

This makes sense, but is it legit? Because the experimenter asked for ranks, (not ratings that were then converted to ranks), ties are not allowed. Also there is concern about non-independence within the block (i.e. if I know three of a subject's ratings, I can deduce the fourth).