Stem-supported Ramsey family need not be completely Ramsey

Let E be the even positive integers, and take a finite-symmetric-difference parity colouring $\epsilon$ of $[E{]}^{\omega }$. The family $\{\{1\}\cup X:X\in [E{]}^{\omega },\epsilon (X)=0\}$ is Ramsey even under the every-ground-set convention, since every infinite ground set can be thinned to omit 1. It is not completely Ramsey: inside $[\{1\},E]$, every infinite tail has two subfamilies of opposite parity. Its supporting neighbourhood is closed and nowhere dense in the ordinary topology on infinite subsets, but open in the Ellentuck topology.