About Bernstein sets

Question

Remember that a subset $X\subseteq \mathbb{R}$ is a $G_{\delta}$-set if $X$ is a countable intersection of open sets in $\mathbb{R}$. For example closed subsets of $\mathbb{R}$ are $G_{\delta}$-sets. A set $Y\subseteq \mathbb{R}$ is dense-in-itself if every point of $Y$ is a limit point of $Y$.

Also a set $B\subseteq \mathbb{R}$ is called a Bernstein set if $P\cap B\not=\emptyset $ and $P \cap (\mathbb{R}\setminus B)\not=\emptyset$ for every uncountable closed set $P\subseteq \mathbb{R}$. Note that if $B$ is a Bernstein set then $\mathbb{R}\setminus B$ is also a Bernstein set.

Let $B\subseteq \mathbb{R}$ be a Bernstein set we have the following properties:

1. $|B|=\mathfrak{c}$
2. $B$ has no isolated points
3. $B$ is dense
4. $B$ is a Baire space
5. If $G\subseteq \mathbb{R}$ is a dense $G_{\delta}$ set, then $B\cap G$ is dense in $B$ and $(\mathbb{R}\setminus B)\cap G$ is dense in $\mathbb{R}\setminus B$.
6. Also if $G\subseteq\mathbb{R}$ is a dense-in-itself $G_{\delta}$ set, then $G\cap B\not=\emptyset$ and $G\cap (\mathbb{R}\setminus B)\not=\emptyset$

My question is this: Someone knows more examples of subsets of the real line that satisfy properties 5 or 6.

Thanks