Would this suffice?
git whatchanged -M5 --summary | grep rename | grep '=>'
Here is a modified version which will do renamed and deleted files:
git whatchanged -M5 --summary | grep -E 'rename.*=>|delete mode'
This will give you all renames from the HEAD of your current branch and it's ancestry including merged parents up to the very first commit. The -M5 will have files that are similar by 50% or more reported as a rename; this may be to low of a percentage but you can change it (The 5 is read as .5, or 50% so you could change it to M8 for 80%). Be warned, it will take a long time if there are a lot of commits.
I suggest you limit the range of commits such as:
git whatchanged -M5 --summary <commit-id>..HEAD | grep rename | grep '=>'
As far as I can tell you will need to start with a commit, I am not sure how you could get a comprehensive list of renamed files across all branches and tags at once. If you have divergent branches you want to check, or branches with independent commit histories in a single repo, you will need to run the suggested command on each branch.