Let $X$ be a compact connected Riemann surface. Letting $\Omega^1(X)$ be the set of holomorphic $1$-forms on $X$, we have a natural map $\Omega^1(X) \to H^1(X, \mathbb{C}) =H_1(X, \mathbb{C})^*$ given by integration along cycles, i.e. a form $\omega$ is sent to the functional sending a cycle $\gamma$ to $\int_\gamma \omega$, extended by $\mathbb{C}$-linearity. Likewise we have a map $\overline{\Omega}(X) \to H^1(X, \mathbb{C})$, where $\overline{\Omega}(X)$ is the set of antiholomorphic $1$-forms. From Hodge theory, the images of these two maps give a direct sum decomposition of $H^1(X, \mathbb{C})$.
Using these facts, I am trying to show that image of the map $H_1(X, \mathbb{Z}) \to \Omega^1(C)^*$, again given by integration along cycles, is a (full rank) lattice. That is, viewing $\Omega^1(C)^*$ as an $\mathbb{R}$-vector space, the induced map $H_1(X, \mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{R} \to \Omega(C)^*$ is an isomorphism. The universal coefficient theorem shows that $H_1(X, \mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{R} \simeq H_1(X, \mathbb{R})$, and I believe that the corresponding map $H_1(X, \mathbb{R}) \to \Omega(C)^*$ is dual to the map $\Omega(C) \to H^1(X, \mathbb{R})$ given by integration along cycles. But I am stuck here, and there seems to already be a problem with my interpretation: functionals in $H^1(X, \mathbb{R})$ ought to be $\mathbb{R}$-valued, but integrating an arbitrary holomorphic $1$-form along a cycle might give a complex number (e.g. integrating the form $i \; dz$ along one of the fundamental loops on an elliptic curve). My interpretation also runs into the following issue: if I make the corresponding interpretation for antiholomorphic forms and assume that $\overline{\Omega}(C) \to H^1(X, \mathbb{R})$ is also an isomorphism, which would be consistent with the result of the exercise, then I conclude that $\Omega(C)$ and $\overline{\Omega}(X)$ have the same image $H^1(X, \mathbb{R}) \subset H^1(X, \mathbb{C})$, contradicting Hodge theory.
What's going wrong here? Am I making incorrect identifications, especially when base changing? The source of this exercise is Brian Conrad's notes on abelian varieties.