Fill in lune of Coxeter Complex

Question

Ultimately I'm trying to draw the tiling of a sphere with triangle group (2,3,4) (i.e., Coxeter complex B3) and shade in various lunes. I figured out how to draw the tiling of the sphere using the blocsphere package (as shown with code below). But I can't for the life of me figure out how to now fill the area between two great circles using blocsphere. Any help with trying to code both the Coxeter complex as well as a lune would be appreciated Here's the code that I have for the tiling of the sphere:

\documentclass{amsart}
\usepackage{blochsphere}
\pgfsetlayers{back,main,front}

\begin{document}


\begin{tikzpicture}
    \begin{scope}
        \begin{blochsphere}[radius=1.5 cm,tilt=25,rotation=-20,nested=true]
            % The next line is automatically included when not nesting
            % but needs to be here when nesting
            \drawBall

            \drawBallGrid[style={draw,opacity=1}]{360}{45}

            \drawGreatCircle{45}{0}%[style={dashed}]
            \drawGreatCircle{-45}{0}%[style={dashed}]
            \drawGreatCircle{45}{-90}
            \drawGreatCircle{-45}{-90}
            
        \end{blochsphere}

    \end{scope}
\end{tikzpicture}


\end{document}

And here is what I'm trying to draw at the end of the day:

Answer 1

Maybe this is not the most straight-forward way, but with the use of the fillbetween library from the pgfplots package you could do like this:

\documentclass[border=10pt]{standalone}
\usepackage{blochsphere}
\pgfsetlayers{back, main, front}

\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usepgfplotslibrary{fillbetween}

\begin{document}

\begin{tikzpicture}
    \begin{blochsphere}[radius=1.5cm, tilt=25, rotation=-20, nested=true]
        \drawBall

        \drawBallGrid[style={draw}]{360}{45}

        \drawGreatCircle{45}{0}
        \drawGreatCircle{-45}{0}
        \drawGreatCircle{45}{-90}
        \drawGreatCircle{-45}{-90}

        \drawGreatCircle[style={red, name path global=A}]{90}{135}
        s\drawGreatCircle[style={red, name path global=B}]{90}{180}
    \end{blochsphere}

    \fill[cyan, opacity=0.5,
        intersection segments={of=A and B, sequence={L1 -- R1[reverse]}}] to[bend right=5] cycle;

\end{tikzpicture}

\end{document}

(The bottom part is not super exact, but I opted for this solution instead of calculating the intersections of the large circles with the circumcircle.)

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