Existence of $V \subset \mathbb{R}^n : V, V^\perp \cap \mathbb{R}_{\geq0}^n = 0$

Suppose $V\cap \mathbb{R}^n_{\geq 0} = \{0\}$, and let $P$ denote orthogonal projection onto $V$. A vector $w$ is in $V^\perp$ iff $P(w)=0$.

Let $H$ be the convex hull of $\{e_1,e_2,\dots,e_n\}$, where $e_i$ is a standard basis vector. Note that $H\subset \mathbb{R}^n_{\geq 0}$. The projection $P(H)$ is a convex subset of $V$. For it to not contain $0$, there must be a hyperplane separating $0$ and $P(H)$, which means there exists some non-zero $v\in V$ such that $\forall(w\in P(H))\: w\cdot v > 0$.

But the condition that $V\cap \mathbb{R}^n_{\geq 0} = 0 $ means that for any non-zero $v\in V$ the components of $v$ cannot all be non-negative (nor all non-positive, else $-v\in\mathbb{R}^n_{\geq 0}$). Hence for any $v\in V$ there is some $e_i$ such that $v\cdot e_i < 0$. Thus $v\cdot P(e_i) = v\cdot e_i < 0$, and since $P(e_i)\in P(H)$, so there cannot be a hyperplane separating $0$ and $P(H)$.

Hence $0\in P(H)$, and thus $H\subset \mathbb{R}^n_{\geq}$ intersects $V^\perp$.

$\newcommand{\IR}{\mathbb{R}}$ Note that $C:=\IR_{\geq 0}^n$ is a cone, i.e. a subset that is closed under addition and multiplication with $\IR_{\geq 0}$. Following Paul Garrett's advice, we look at the dual cone, i.e. $C^\vee := \{v \mid \forall x\in C: \langle v,x\rangle \geq 0\}$. One can easily verify that $V^\vee = V^\perp$ for subspaces $V\leq\IR^n$ and $(\IR_{\geq 0}^n)^\vee = \IR_{\geq 0}^n$.

Now look at any basis $w_1,\ldots,w_k$ of $V^\perp$ and look at the linear map $\omega: \IR^n\to\IR^k, x\mapsto (\langle w_1,x\rangle, ..., \langle w_k,x\rangle)$. By assumption $\ker(\omega)=V$ so that $\omega(\IR_{\geq0}^n\setminus\{0\}) \subseteq \IR^k\setminus\{0\}$. This is now a cone in $\IR^k$ and by choosing a different basis of $\IR^k$ (or equivalently: choosing a different basis of $V^\perp$), we can assume that $\omega(\IR_{\geq0}^n) \subseteq \IR_{\geq0}^k$. But that means that $\langle w_i,x\rangle \geq 0$ for all $x\in C$, i.e. $w_i\in (\IR_{\geq 0}^n)^\vee = \IR_{\geq 0}^n$.

We have shown: If $V\cap\IR_{\geq 0}^n=0$, then $V^\perp\cap\IR_{\geq 0}^n$ contains a whole basis of $V^\perp$.