[Electronic M]

That stuff is called aquadag (conductive silver paint), and it's purpose is to filter the HV. The outer coating is grounded (or should be) and the inner coating (inside the CRT glass walls) is connected to the HV....Together with the glass between them they form a capacitor that filters the HV. If that coating is not or poorly grounded many strange symptoms can result, including (in the worst case) CRT implosion (a member reported having a bad arcing ground connection that caused implosion after a few minutes of arcing).
If patches of the grounded side are disconnected from the rest arcing may occur. If the grounded side is disconnected it may float up to HV potential and shock you while servicing the rest of the set.

It likes to break into small bits once it de-laminates so it would be a good idea to clean all the bits and any on the tube that are loose...Unless you are fine with the risk of it shorting and burning up one of the more delicate not necessarily easy or cheap to replace coils (IF, h osc, traps, etc.).....