Husk, 10 bytes

ωoṁS↑o%2Lg

Try it online! Explanation to come later.

Husk, 10 bytes

ωoṁS↑o%2Lg

Try it online!

Explanation

ωoṁS↑o%2Lg
ω           Repeat until fixed point
 o          the following two functions:
         g   a) group adjacent elements
  ṁ          b) map over groups and concatenate:
        L     length of group
     o%2      mod 2
   S↑         take that many elements of group