Let $F=\mathbb{R}(t)$ be the field of rational functions in the variable $t$ with real coefficients. We regard $F$ as a structure of type $(+ \times -\,\, 0\,\, 1)$. Then

The (positive) existential theory of $F$ is effectively computable (e.c.)

The full first-order theory of $F$ is not e.c.

Proof of 1: Suppose that a system of polynomial equations has a solution $\bar{r}$ in $F$. Here $\bar{r}$ is a tuple of rational functions. Choose a real number $s$ that is not a root of any of the denominators of the rational functions $r_i$ and substitute $s$ for $t$ in $\bar{r}$, to obtain a tuple of real numbers that satisfies the same system of equations. Conversely, a tuple of reals that satisfies a given system of equations is already a tuple of rational functions. It follows that the existential theory of $F$ is e.c. if and only if the existential theory of $\mathbb{R}$ is e.c. But the last statement is true, by a well-known theorem of Tarski.

Proof of 2: A proof of the undecidablity of the first-order theory of $F$ (actually $\mathbb{R}$ can be replaced by any archimedian formally real field) is the subject of a 1961 paper by Raphael Robinson here. Especially, look at Section 3, "The Method of Julia Robinson." The argument shows (amazingly) that the natural numbers can be defined in $F$.