Blind grading for exams isn't very blind TL;DR:

How should I blindly grade exams when I can recognize over half of the papers by the handwriting?

Situation: I'm a TA for an first year physics class of approximately 50-60 students. Due to a complaint along the lines of "the grader hates me so I failed the class" that the department got last year, there is a proposal in the department to ensure that tests be swapped to blind grading. However, as the class assigns a fairly large amount of homework, from experience I know that I can reliably recognize more than half of the class's handwriting. The proposed system goes along the lines of having students use an ID number that they get when they take the test that isn't shown to me until I'm putting grades into the computer. The problem is that I feel that this would do very little to actually reduce bias if TAs can just recognize handwriting without a name.

Short of putting people on rotation for grading this stuff, which has its own problems, are there any better ways to implement a blind grading system to remove any handwriting bias?
If I understand this correctly, blind grading is a formality introduced by the University to make students' claims on the biased teachers unjustified. I would say - keep it that way and do not bother any more. You are not going to hate them and give them lower grades, are you? So, it does not really matter that you in fact can recognise the handwriting. If a student is really really bothered to keep their identity secret... well, they could try to write with the unusual hand.

Answering your question formally - to implement a real blind grading system you can swap the scripts between departments, or TAs.