right)^-$.
\item[3.] $\flag$ is a $s$-flag of flats of $M$.
\end{enumerate}
\end{theorem}
\begin{proof}
At first, we set $A=s^-$, i.e. we reorientate with respect to $A$. Note that reorienting and initializing a matroid $M$ commutes, i.e. for $A\subseteq E$ and $\flag=\flag(w)$ for $w\in\R^n$ we have
\begin{equation}\notag
{}_{-A}\left( M_\flag \right) = \left( {}_{-A}M \right)_\flag.
\end{equation}
The reason for this is that $w$ picks the elements of a (signed) circuit $C$ where $w$ is maximal independently from the signs. Circuits of the underlying matroid $\umatroid{M}$ remain invariant under reorientation. Also note that covectors $v\in\cvector{M}$ translate to covectors $s\cdot v\in\svector_{{}_{-A}M}$ of ${}_{-A}M$ (\cite[Section 3 + Lemma 4.18]{bjoernerlasvergnassturmfelswhiteziegler}) and finally, ${}_{-A}\left( {}_{-A}M \right)=M$ and $s\cdot \left( s\cdot v \right)=v$.\\
For $1.\Rightarrow 2.$ suppose $M_\flag$ is $s$-acyclic. Thus, $\reorientedmatroid{A}{\left(M_\flag\right)}=\left(\reorientedmatroid{A}{M}\right)_\flag$ is acyclic (cf. \Cref{lemm:acyclic}) and we have $s(\initialcircuit{w}{\reoriented{A}{C}})\nsubseteq(+)^n$ for all circuits $C\in\circuits$. Since reorientation commutes with initialization, we have $\reoriented{A}{s}\cdot s(\initialcircuit{w}{C})\nsubseteq(+)^n$ for all $C\in\circuits$. Consequently, for all $C\in\circuits$ exist $e,f\in E$ such that (w.l.o.g.) $(s)_e (s(\initialcircuit{w}{C}))_e = -$ and $(s)_f (s(\initialcircuit{w}{C}))_f = +$. Since $\initialcircuit{w}{C}\subseteq C$ we conclude that for all $C\in\circuits$ exist $e,f\in E$ such that $(s)_e (s(C))_e=-$ and $(s)_f (s(C))_f =+$. In other words, for all $C\in\circuits$ holds that $w$ attains its maximum at $(s\cdot s(C))^+$ and $(s\cdot s(C))^-$.\\
Vice versa, suppose that $w$ attains its maximum at $(s\cdot s(C))^+$ and $(s\cdot s(C))^-$ for all $C\in\circuits$. Since $s\cdot s_C =\reoriented{A}{s}\cdot s_C$ we conclude that $(\reoriented{A}{s}\cdot s(\initialcircuit{w}{C}))^\pm\neq\emptyset$ for all $C\in\circuits$. Hence, $s\cdot s(\initialcircuit{w}{C}) = s(\initialcircuit{w}{\reoriented{A}{C}})\nsubseteq(+)^n$ for all $C\in\circuits$, i.e. $\left(\reorientedmatroid{A}{M}\right)_\flag$ is acyclic. \Cref{lemm:acyclic} implies that $M_\flag$ is $s$-acyclic.\\
For $1.\Rightarrow 3.$ suppose that $M_\flag$ is $s$-acyclic. Hence, $\left(\reorientedmatroid{A}{M}\right)_\flag$ is acyclic (\Cref{lemm:acyclic}). Thus, $\flag$ is a positive flag of flats of $\reorientedmatroid{A}{M}$ (\Cref{theo:positivebergmanfan}). Let $\flag=(F_1,\ldots,F_l)$ be the flag of flats and $\{v_1,\ldots,v_l\}\subset\cvector{\reorientedmatroid{A}{M}}$ the covectors such that $v_i^0=F_i$ for $1\leq i \leq l$. According to \Cref{rema:reorientation}, $\{s\cdot v_1,\ldots,s\cdot v_l\}\subset\cvector{M}$ is a set of covectors satisfying $(s\cdot v_i)^0=F_i$ for $1\leq i\leq l$. To see this note that $s$ is a pure sign vector, i.e. $s^0=\emptyset$. Thus, $(s\cdot v_i)_e=0$ if and only if $(v_i)_e=0$. Moreover, $v_i\subset(+)^n$, i.e. $v_i^-=\emptyset$ for $1\leq i\leq l$. Thus, $s_i=(s\cdot v_i)\subseteq s$ and, therefore, $\{s_1,\ldots,s_l\}=\{s\cdot v_1,\ldots,s\cdot v_l\}\subset\cvector{M}$ is a set of covectors such that $\flag$ is a $s$-flag.\\
Vice versa, let $\flag$ be a $s$-flag, i.e. $\flag=(F_1,\ldots,F_l)$ and there is a set $\{v_1,\ldots,v_l\}\subset\cvector{M}$ such that $F_i=v_i^0$ and $v_i\subseteq s$. Using \Cref{rema:reorientation} again, we conclude that $\{s\cdot v_1,\ldots,s\cdot v_l\}=\{s_1,\ldots,s_l\}\subset\cvector{\reorientedmatroid{A}{M}}$ is a set of covectors such that $s_i^0=(s\cdot v_i)^0 =v_i^0$ and $s_i\subset (+)^n$ for $1\leq i\leq l$. Hence, $\flag$ is a positive flag of $\reorientedmatroid{A}{M}$. Due to \Cref{theo:positivebergmanfan} this implies $\left(\reorientedmatroid{A}{M}\right)_\flag$ is acyclic. By \Cref{lemm:acyclic} this is equivalent to $M_\flag$ is $s$-acyclic.
\end{proof}
\begin{corollary}
Let $M$ be an oriented matroid and $\topes$ its set of topes (cf. \Cref{rema:topes}). Then:
\begin{equation}\notag
s\notin\topes\quad\Rightarrow\quad\signedbergmanfan{s}{M}=\emptyset.
\end{equation}
\end{corollary}
\begin{proof}
Assume $\signedbergmanfan{s}{M}\neq\emptyset$. Then, there is an element $w\in\signedbergmanfan{s}{M}$ such that $M_w$ is $s$-acyclic. By \Cref{theo:signedbergmanfan} we know that $\flag\left( w \right)$ is a $s$-flag of $M$. Hence, $s\in\topes$. 
\end{proof}

From \Cref{theo:signedbergmanfan} we can immediately deduce the following
\begin{corollary}
Let $M$ be an oriented matroid. Then, the signed Bergman fan of $M$ with respect to $s\in\pvector{}$ is the union of weight class defined by $s$-flags of flats of $M$:
\begin{equation}\notag
\signedbergmanfan{s}{M}=\bigcup_{\substack{\flag\triangleleft\umatroid{M}:\\\flag~\text{is an}~s\text{-flag}}}\sigma_\flag.
\end{equation}
\end{corollary}

Let us turn back to real tropicalizations of real linear spaces over $\RealField$. The positive part of $\realtrop\left(\Var{\Ideal{I}}\right)$ is well-understood (\cite[Proposition 4.1]{ardilaklivanswilliams}). For $\Ideal{I}\subset\RealLauring{n}$ a linear ideal with constant coefficients and $M$ the associated oriented matroid we have $\realtrop\left(\Var{\Ideal{I}}\right)\cap\left( (+)^n\times\R^n \right) = \signedbergmanfan{+}{M}$. We generalize this statement for arbitrary pure sign vectors $s\in\svector$:
\begin{theorem}\label{theo:realtropicallinearspace=signedbergmanfan}
Let $\Ideal{I}\subset\