 Solution

Take the cone on an infinite coinfinite ground setIt is $\ast$-open because $E=[\varnothing ,2\mathbb{N}]$ is a basic open set of the Ellentuck topology. It is not $\tau$-open: if $X\in E$, every ordinary basic neighbourhood $[s]$ of $X$ contains an infinite extension of $s$ having an odd point larger than $\max s$. That extension is outside $E$. Therefore no point of $E$ has an ordinary open neighbourhood contained in $E$.