  With the aim of providing a worldsheet description of the refined topological
string, we continue the study of a particular class of higher derivative
couplings $F_{g,n}$ in the type II string effective action compactified on a
Calabi-Yau threefold. We analyse first order differential equations in the
anti-holomorphic moduli of the theory, which relate the $F_{g,n}$ to other
component couplings. From the point of view of the topological theory, these
equations describe the contribution of non-physical states to twisted
correlation functions and encode an obstruction for interpreting the $F_{g,n}$
as the free energy of the refined topological string theory. We investigate
possibilities of lifting this obstruction by formulating conditions on the
moduli dependence under which the differential equations simplify and take the
form of generalised holomorphic anomaly equations. We further test this
approach against explicit calculations in the dual heterotic theory.
