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Maybe you should try without list_plot?

(no random points)

Replace li by your list of [x,y(x),ymax(x),ymin(x)]

li=[[x,sin(x),sin(x)+0.2,sin(x)-0.2] for x in srange(0,2*pi,0.3)]

a=[[x[0],x[2]] for x in li]

b=[[x[0],x[3]] for x in li]

c=[[x[0],x[1]] for x in li]

b.reverse()

p1=polygon2d(a+b,color='lightgrey',alpha=0.7)

p2=line(c,thickness=3)

p3=line(a,color='black')

p4=line(b,color='black')

(p1+p2+p3+p4).show()