Filling a region in Tikz

\documentclass[tikz, border=1cm]{standalone}
\begin{document}
\begin{tikzpicture}
\begin{scope}
  \clip (0,0)--(12,6)--(12,0)--cycle; 
  \clip (3,0) rectangle (12,6);
  \fill[fill=red] (0,0) rectangle (12,6) (3,3) circle[radius=3cm] (9,3) circle[radius=3cm];
\end{scope}
\draw (0, 0) rectangle (12, 6);
\draw (3,3) circle [radius=3cm];
\draw (9,3) circle [radius=3cm];
\draw (0, 0)--(12,6);
\end{tikzpicture} 
\end{document}

Not the most elegant solution as you have to adjust your clipping area, when moving the circles around, but I'm sure this could be refined (e.g. using the calc library).

\documentclass[tikz, border=1cm]{standalone}
\begin{document}

    \begin{tikzpicture}
        \draw (0, 0) rectangle (12, 6);
        \draw (3,3) circle [radius=3cm];
        \draw (9,3) circle [radius=3cm];
        \draw (0, 0)--(12,6);
        \begin{scope}[even odd rule]
            % you have to adjust the following line when moving your circles around
            \clip  (3,0)    % bottom of first circle
                -- (6,3)    % intersection of circles
                -- (12,6)   % upper right corner
                -- (12,0)   % lower right corner
                -- cycle;
            \fill (3,3) circle [radius=3cm] (0, 0) -- (12,6) -- (12, 0) -- cycle (9,3) circle [radius=3cm];
        \end{scope}

    \end{tikzpicture}

\end{document}

very close:

\documentclass[tikz, border=1cm]{standalone}
\begin{document}

\begin{tikzpicture}
\draw[fill=black] (0,0) rectangle (12,6);
\draw[fill=white] (0,0) -- (12,6) -| cycle;
\draw[fill=white] (3,3) circle [radius=3cm];
\draw[fill=white] (9,3) circle [radius=3cm];
\draw (0, 0)--(12,6);
\end{tikzpicture}

\end{document}

more close:

\documentclass[tikz, border=1cm]{standalone}
\begin{document}

\begin{tikzpicture}
\draw[fill=black] (0,0) rectangle (12,6);
\fill[white] (0,0) -- (3,0) -- (3,3) -- cycle;
\draw[fill=white] (0,0) -- (12,6) -| cycle;
\draw[fill=white] (3,3) circle [radius=3cm];
\draw[fill=white] (9,3) circle [radius=3cm];
\draw (0, 0)--(12,6);
\end{tikzpicture}

\end{document}