I can't think of any example of a topological space satisfying the Baire Category Theorem which is not Cech-complete. (A space is Cech-complete if it is a Tychonoff space and if it is a $G_\delta$ subset of its Stone-Cech compactification, or equivalently of any other compactification). There isn't even an example of one in $\pi$-base. However, it's very well known every Cech-complete space is Baire, and I've never seen it mentioned that the reverse is true as well.
Are these properties equivalent? Is there a counterexample?
The space $\mathbb{N}^{\omega_1}$ is Baire (as any product of Čech-complete spaces is, Engelking 3.9.J(e), due to Oxtoby or Bourbaki), but it is not a $k$-space (see Engelking 3.3.E(a)) so not Cech-complete.
A metrisable example: $X = \mathbb{Q}\times \{0\} \cup \mathbb{R} \times (0,\infty) \subseteq \mathbb{R}^2$ (subspace topology) is Baire as $\mathbb{R} \times (0,\infty)$ is open and dense and Baire (as a locally compact space, e.g.). But $X$ cannot be Čech-complete as the closed subset $\mathbb{Q} \times \{0\} \simeq \mathbb{Q}$ would be Čech-complete (the class is closed under closed subsets and $G_\delta$ subsets) which it is not.