The white stringy slime you are seeing is actually the secretion from a sulfur feeding bacterium. The sulfur in your fertilizer is providing sustenance for the bacterium which is producing the slime. Unfortunately sulfur is necessary in fertilizer to counteract the alkalinity of salts. However, I am not sure of the exact sulfur content in Fertigators fertilizers, but their has to be some. If you discontinued the use of the Fertigator for an extended period of time, the problem would disappear assuming there is no other sulfur source.
There are two treatments. The one you use depends on the water source for your irrigation system.
Well water- you have to treat the well itself because this is where the bacterium is originating. The bacterium has been in your well for quite some time, and the Fertigator just allowed it to become visible. The solution to the problem is chlorine. You can shock the system, very similar to a shock treatment of a pool. The chlorine will kill the bacteria and inhibit future growth. Sulfur feeding bacteria is very resistant to chlorine because of the thick layers of slime (white stringy stuff). The secretion is what protects the bacteria. A standard treatment may kill off the bacteria on the surface layers of the slime but will leave the inner layers unaffected and free to continue multiplying. I would strongly consider consulting a well specialist to perform the service for you. It may require multiple chlorine treatments before the problem is controlled.
City water- it is rare that city water will carry this bacterium due to the high levels of chlorine injected at the water treatment plant. However, if this is the case, it is more difficult to fix. You will need to introduce chlorine into your irrigation lines on a periodic basis. If the bacteria only exists in your irrigation lines it will eventually go away completely after multiple treatments. The hard part is introducing the chlorine into your irrigation line. Do not worry it can be done, but I can provide instructions if you need me to. My feeling is you are using a well...