Python 2, 149 bytes

a=input() l=len(a) n=p=0 for i in range(l): for j in range(l-1,i+3,-1): if(j>p)>(.15<sum(a[i:j+1])/(j+1.-i)+a[i]+a[j]<2.85):n+=1;p=j;break print n 

This sweeps across the array, checking if each bit can be the first in a maximal phase.

Example:

$ python phase.py [1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0] 3 

Python 2, 149 bytes

a=input() l=len(a) n=p=0 for i in range(l): for j in range(l-1,i+3,-1): if(j>p)>(.15<sum(a[i:j+1])/(j+1.-i)+a[i]+a[j]<2.85):n+=1;p=j;break print n 

The first loop scans across the array from left to right. Each bit, indexed by i, is checked to see if it could be the first bit in a maximal phase.

This is done by the inner loop, which scans from right to left. If the subarray between i and j is a phase, we increase the counter and move on. Otherwise, we keep going until the subarray becomes too small or j reaches the end of the previous maximal phase.

1 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 i -> <- j 

Example:

$ python phase.py [1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0] 3