Yes, it's possible:
The code:
\documentclass{article}
\usepackage{tikz-cd}
\definecolor{cof}{RGB}{219,144,71}
\definecolor{pur}{RGB}{186,146,162}
\definecolor{greeo}{RGB}{91,173,69}
\definecolor{greet}{RGB}{52,111,72}
\begin{document}
\begin{tikzcd}[execute at end picture={
\foreach \Valor/\Nombre in
{%
tikz@f@1-2-1/a,tikz@f@1-1-2/b,tikz@f@1-1-4/c,tikz@f@1-2-3/d,%
tikz@f@1-4-1/e,tikz@f@1-3-2/f,tikz@f@1-3-4/g,tikz@f@1-4-3/h%
}
{
\coordinate (\Nombre) at (\Valor);
}
\fill[pur,opacity=0.3]
(a) -- (b) -- (c) -- (d) -- cycle;
\fill[cof,opacity=0.3]
(a) -- (d) -- (h) -- (e) -- cycle;
\fill[greeo,opacity=0.3]
(h) -- (d) -- (c) -- (g) -- cycle;
}
]
&
P
\ar{dl}[swap, sloped, near start]{t}
\ar{rr}{t'}
\ar[]{dd}[near end]{t}
& & R''
\ar{dd}{}
\ar{dl}[swap, sloped, near start]{}
\\
R
\ar[crossing over]{rr}[near start]{}
\ar{dd}[swap]{}
& & S''
\\
&
R'
\ar[near start]{rr}{}
\ar[sloped, near end]{dl}{}
& & S
\ar{dl}
\\
S'
\ar{rr}
& & P'
\ar[crossing over, leftarrow, near start]{uu}{}
\end{tikzcd}
\end{document}
The idea is to place a coordinate at each of the nodes in the diagram and then use those coordinates to do the filling; this is done using the execute at end picture key.
The internal numbering schema
Each node in a tikz-cd diagram has an internal name of the form
tikz@f@<number>-<row>-<column>
where <number> is the number of the tikz-cd diagram, <row> and <column> are the row and column, respectively, of the node. For example, the node labeled R in the above diagram (assuming it is the first tikz-cd diagram) would be
tikz@f@1-2-1
Had it been the 13th diagram, the internal name for R would've been
tikz@f@13-2-1
For example, the following shows, for each node, both its internal name and its new user-defined name. This is helpful if you want some guide during the filling process:
\documentclass{article}
\usepackage{tikz-cd}
\definecolor{cof}{RGB}{219,144,71}
\definecolor{pur}{RGB}{186,146,162}
\definecolor{greeo}{RGB}{91,173,69}
\definecolor{greet}{RGB}{52,111,72}
\begin{document}
\begin{tikzcd}[execute at end picture={
\foreach \Valor/\Nombre/\Posi in
{%
tikz@f@1-2-1/a/left,tikz@f@1-1-2/b/left,tikz@f@1-1-4/c/right,tikz@f@1-2-3/d/right,%
tikz@f@1-4-1/e/left,tikz@f@1-3-2/f/left,tikz@f@1-3-4/g/right,tikz@f@1-4-3/h/right%
}
{
\coordinate (\Nombre) at (\Valor);
\node[\Posi=10pt,align=center] at (\Nombre) {\Valor\\ new name:\Nombre};
}
}
]
&
P
\ar{dl}[swap, sloped, near start]{t}
\ar{rr}{t'}
\ar[]{dd}[near end]{t}
& & R''
\ar{dd}{}
\ar{dl}[swap, sloped, near start]{}
\\
R
\ar[crossing over]{rr}[near start]{}
\ar{dd}[swap]{}
& & S''
\\
&
R'
\ar[near start]{rr}{}
\ar[sloped, near end]{dl}{}
& & S
\ar{dl}
\\
S'
\ar{rr}
& & P'
\ar[crossing over, leftarrow, near start]{uu}{}
\end{tikzcd}
\end{document}
If you don't want to keep finding the currect number of your diagram, you can use the associated internal count(er) \tikz@fig@count, but then something like
tikz@f@1-2-1
transforms into
tikz@f@\the\tikz@fig@count-2-1
and the resulting code is rather cumbersome:
\makeatletter
\begin{tikzcd}[
execute at end picture={
\foreach \Valor/\Nombre in
{%
tikz@f@\the\tikz@fig@count-2-1/a,tikz@f@\the\tikz@fig@count-1-2/b,tikz@f@\the\tikz@fig@count-1-4/c,tikz@f@\the\tikz@fig@count-2-3/d,%
tikz@f@\the\tikz@fig@count-4-1/e,tikz@f@\the\tikz@fig@count-3-2/f,tikz@f@\the\tikz@fig@count-3-4/g,tikz@f@\the\tikz@fig@count-4-3/h%
}
{
\coordinate (\Nombre) at (\Valor);
}
\fill[cyan,opacity=0.15]
(a) -- (b) -- (c) -- (d) -- cycle;
\fill[orange,opacity=0.2]
(e) -- (f) -- (g) -- (h) -- cycle;
\fill[green,opacity=0.12]
(e) -- (a) -- (d) -- (h) -- cycle;
}
]
&
P
\ar{dl}[swap, sloped, near start]{t}
\ar{rr}{t'}
\ar[]{dd}[near end]{t}
& & R''
\ar{dd}{}
\ar{dl}[swap, sloped, near start]{}
\\
R
\ar[crossing over]{rr}[near start]{}
\ar{dd}[swap]{}
& & S''
\\
&
R'
\ar[near start]{rr}{}
\ar[sloped, near end]{dl}{}
& & S
\ar{dl}
\\
S'
\ar{rr}
& & P'
\ar[crossing over, leftarrow, near start]{uu}{}
\end{tikzcd}
\makeatother
A better approach
Using the alias key (and the standard |[...]| syntax for passing keys to nodes in a matrix), you can place easy-to-remember names to the nodes so you don't have to fiddle with the internal names and can use those simpler names for the shading:
\documentclass{article}
\usepackage{tikz-cd}
\definecolor{cof}{RGB}{219,144,71}
\definecolor{pur}{RGB}{186,146,162}
\definecolor{greeo}{RGB}{91,173,69}
\definecolor{greet}{RGB}{52,111,72}
\begin{document}
\begin{tikzcd}[execute at end picture={
\foreach \Nombre in {A,B,...,H}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[pur,opacity=0.3]
(A) -- (B) -- (C) -- (D) -- cycle;
\fill[greeo,opacity=0.3]
(A) -- (D) -- (H) -- (E) -- cycle;
\fill[cof,opacity=0.3]
(H) -- (D) -- (C) -- (G) -- cycle;
}]
&
|[alias=B]|P
\ar{dl}[swap, sloped, near start]{t}
\ar{rr}{t'}
\ar[]{dd}[near end]{t}
& & |[alias=C]|R''
\ar{dd}{}
\ar{dl}[swap, sloped, near start]{}
\\
|[alias=A]|R
\ar[crossing over]{rr}[near start]{}
\ar{dd}[swap]{}
& & |[alias=D]|S''
\\
&
|[alias=F]|R'
\ar[near start]{rr}{}
\ar[sloped, near end]{dl}{}
& & |[alias=G]|S
\ar{dl}
\\
|[alias=E]|S'
\ar{rr}
& & |[alias=H]|P'
\ar[crossing over, leftarrow, near start]{uu}{}
\end{tikzcd}
\end{document}
Use of backgrounds
For diagrams in which crossing over or description are not used, one can refine the answer doing the filling in a background layer; in this way, even with opacity=1 the filling doesn't affect labels nor arrows:
\documentclass{article}
\usepackage{tikz-cd}
\pgfdeclarelayer{background}
\pgfsetlayers{background,main}
\definecolor{cof}{RGB}{219,144,71}
\definecolor{pur}{RGB}{186,146,162}
\definecolor{greeo}{RGB}{91,173,69}
\definecolor{greet}{RGB}{52,111,72}
\begin{document}
\begin{tikzcd}[
execute at end picture={
\begin{pgfonlayer}{background}
\foreach \Nombre in {A,B,...,E}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[pur!40]
(A) -- (B) -- (D) -- (C) -- cycle;
\end{pgfonlayer}
}
]
|[alias=A]|A
\ar[rr,"f"]\ar[dd,"k"]\ar[dr,"\alpha"]
& &
|[alias=B]|B\ar[dd,"l",swap]
\\
&
|[alias=E]|E\ar[ur,"\beta"]\ar[dr,"\delta"]
&
\\
|[alias=C]|C\ar[rr,"g"]\ar[ur,"\gamma"]
& &
|[alias=D]|D
\\
\end{tikzcd}
\end{document}
The result:
