A space $X$ is called an almost $P$-space if $Int(G) \neq \emptyset$ for every non-empty $G_\delta$ subset $G \subset X$.

Every $P$-space (that is, a space where $G_\delta$s are open) is an almost $P$-space, but the class of almost $P$-spaces is much wider and even contains some compact spaces like $\omega^*$.

In his answer to another question, Joseph van Name notes that no compact group can be an almost $P$-space (this is an easy consequence of the existence of a Haar probability measure). Joseph's observation suggests a natural question:

QUESTION: Is there a compact Hausdorff homogeneous almost $P$-space?

Ronnie Levy notes that every compact linearly ordered almost $P$-space has a $P$-point, hence no example answering the above question can be a linearly ordered space (if $X$ is homogeneous and has a $P$-point then every point of $X$ must be a $P$-point, but every compact space must have a non-$P$-point).