There is small numerical noise along the imaginary axis, which messes up the plot. Just insert Chop to get rid of it:
qlsol[\[Gamma]_?NumericQ, n_?NumericQ, \[Beta]_?NumericQ,
pih_?NumericQ, \[Epsilon]h_?NumericQ, \[Epsilon]l_?
NumericQ, \[Alpha]_?NumericQ, A_?NumericQ, \[Sigma]_?NumericQ] :=
Plot[Chop[-((\[Gamma] - \[Beta] n)/(\[Beta] (1 - n))) +
pih \[Epsilon]h (\[Sigma] du[
qhsol1[\[Gamma], n, \[Beta],
pih, \[Epsilon]h, \[Epsilon]l, \[Alpha], A, ql], \[Alpha],
A] + (1 - \[Sigma]) dg[
qhhatsol[\[Gamma], n, \[Beta],
pih, \[Epsilon]h, \[Epsilon]l, \[Alpha], A, ql], \[Alpha],
A]) + (1 -
pih) \[Epsilon]l ( \[Sigma] du[ql, \[Alpha],
A] + (1 - \[Sigma]) dg[
qlhatsol[\[Gamma], n, \[Beta],
pih, \[Epsilon]h, \[Epsilon]l, \[Alpha], A, ql], \[Alpha],
A])], {ql, 10.9, 11.1}];