Jasper it looks bad, sorry to say.
Your description is of white rot, a fungal disease often found on allotments from gardeners not knowing the plot history, (like you), and over-cultivation of soil.
The photos however show quite healthy roots which would be covered in white fungus if its white rot. I still think you have that, perhaps also some disease brought in with sets or even the garlic seed.
No remedy I know once plants are affected.
Can you make a bed in your back garden, on clean soil?
