Due to the unpredictability of the disease a horse that is positive usually must be put down, even if asymptomatic.
I knew someone who had a lovely Quarterhorse mare. She was EIA Positive (this was when they first started testing). This mare was in a field with another retired mare and a retired gelding. She was a spectacular mare and she was bred to My Bay Bailey. I think this was right around the time the coggins first came out.. so she may have been bred before there was a test.. but then was tested after she was confirmed pregnant per vet recommendations (it was a long time ago).
Anyway, she tested positive and never showed disease symptoms. She foaled a filly (Bailey Bay Sou) and the foal tested positive while nursing but 8 weeks after being weaned tested negative.
The mare was left in the pasture with the two retired horses after that for YEARS. She tested positive and never was sick. Both the horses with her always tested negative. The horses were all put down at very advanced ages for other reasons (one no longer could get up.. he was 30, one had melanomas.. a grey.. and she was in her early 20's and the QH mare who was positive was in her 20's and had ring bone and was lame).