Colored background for \sidebar in Memoir

Question

I am looking to set a background color for my sidebars in the memoir class. Is there a way to do this? I am looking to add a light grey background to my side notes so that they stand out. Other options for making them stand out, beyond a simple background color will also be useful.

\documentclass{memoir}
\begin{document}

\setsidebars{*}{3cm}{*}{*}{*}{*}
\sidebar{ 
\textbf{Tip:} 

\smallskip
Here's a really useful trick to comparing fractions. It only works for two 
fractions at one time, but you can use it a few times if you have a few fractions.

\smallskip
You can cross-multiply and compare the products to see which fraction is larger.
}

\end{document}

Edit: So after putting in your solution, I am having a conflict with tikz boxes I have used. Here is a MWE with the issue. I figured this would be better than a screenshot. The solution is perfect for what I want, but I have a great many of these tikz boxes throughout the document.

\documentclass{memoir}
\usepackage{lipsum}
\usepackage{mdframed}
\usepackage{tikz}

% Sidebar
\makeatletter
\renewcommand{\sidebar}[1]{%
  \insert\sideins{%
\begin{mdframed}[
  backgroundcolor=gray!30,
  skipabove=0pt,
  innertopmargin=0pt,
  userdefinedwidth=\sidebarwidth,
  frametitle=Tip:,
  frametitlebelowskip=0pt]
    \hsize\sidebarwidth
    \@parboxrestore
    \def\baselinestretch{\m@m@footnote@spacing}%
    \m@mwhich@margin{\m@msidebar@margin}%
    \sidebarform\sidebarfont
    \splittopskip=\ht\strutbox
    \splitmaxdepth=\dp\strutbox % doesn't do anything useful
    \allowbreak
    \prevdepth=\dp\strutbox    % supersedes a "top-strut"
    \vskip-\parskip
    \parskip=\smallskipamount% optional
    #1%
    \ifvmode\else
      \unskip\@finalstrut\strutbox
    \fi\par
    \ifdim\prevdepth>\dp\strutbox \prevdepth=\dp\strutbox \fi
    \ifdim\prevdepth>99\p@
      \nobreak
      \vskip-\prevdepth
      \allowbreak
      \vskip\dp\strutbox
    \fi
    \vskip\sidebarvsep\end{mdframed}}%
}
\makeatother
% Tikz
\tikzset{
fancytitle/.style={fill=blue!15,draw=black,thick,text=black,rectangle,rounded corners},
lbox/.style= {draw=black,fill=black!5,thick,rectangle,rounded corners,inner xsep=7pt, inner ysep=15pt},
rbox/.style= {draw=black,fill=white,thick,rectangle,rounded corners,inner xsep=2pt, inner ysep=2pt}
}

\begin{document}


%         Tips Sidebar
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\setsidebars{*}{2cm}{*}{*}{*}{*}
% Parameters set with this command. Any command with `*' denotes no change.
% \setsidebars{⟨hsep⟩}{⟨width⟩}{⟨vsep⟩}{⟨topsep⟩}{⟨font⟩}{⟨height⟩}
\sidebar{ 

Here's a really useful trick to comparing fractions. It only works for two 
fractions at one time, but you can use it a few times if you have a few fractions.

You can cross-multiply and compare the products to see which fraction is larger.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%        
\begin{minipage}{\textwidth}
\begin{tikzpicture}[transform shape]
\node [lbox] (box) {
    \begin{minipage}{.4\textwidth}
        {
        \lipsum[1]
        }
    \end{minipage}
    };
\node[fancytitle,right=8pt] at (box.north west) {Lipsum};
\end{tikzpicture}
\hfill
\begin{tikzpicture}[transform shape]
\node [rbox] (box) {
    \begin{minipage}{.4\textwidth}
      { 
       \lipsum[1]
    }
    \end{minipage}
    };
    \node[fancytitle,right=8pt] at (box.north west) {Lipsum};
\end{tikzpicture}
\end{minipage}


\end{document}

Answer 1

One possibility, using the mdframed package to add a colored frame; of course, using the features provided by the package, you can customize further the sidebars:

\documentclass{memoir}
\usepackage{xcolor}
\usepackage{mdframed}
\usepackage{lipsum}

\makeatletter
\renewcommand{\sidebar}[1]{%
  \insert\sideins{%
\begin{mdframed}[backgroundcolor=gray!30,skipabove=0pt]
    \hsize\sidebarwidth
    \@parboxrestore
    \def\baselinestretch{\m@m@footnote@spacing}%
    \m@mwhich@margin{\m@msidebar@margin}%
    \sidebarform\sidebarfont
    \splittopskip=\ht\strutbox
    \splitmaxdepth=\dp\strutbox % doesn't do anything useful
    \allowbreak
    \prevdepth=\dp\strutbox    % supersedes a "top-strut"
    \vskip-\parskip
    #1%
    \ifvmode\else
      \unskip\@finalstrut\strutbox
    \fi\par
    \ifdim\prevdepth>\dp\strutbox \prevdepth=\dp\strutbox \fi
    \ifdim\prevdepth>99\p@
      \nobreak
      \vskip-\prevdepth
      \allowbreak
      \vskip\dp\strutbox
    \fi
    \vskip\sidebarvsep\end{mdframed}}%
}
\makeatother

\begin{document}

\setsidebars{*}{3cm}{*}{*}{*}{*}
\lipsum[1-3]\sidebar{% 
\textbf{Tip:} 
\par\smallskip
Here's a really useful trick to comparing fractions. It only works for two 
fractions at one time, but you can use it a few times if you have a few fractions.
\par\smallskip
You can cross-multiply and compare the products to see which fraction is larger.}

\end{document}

Another advantage is that now you can automatically set the title "Tip:" and also set \parskip to be locally \smallskipamount, so you save a lot of manual work:

\documentclass{memoir}
\usepackage{xcolor}
\usepackage{mdframed}
\usepackage{lipsum}

\makeatletter
\renewcommand{\sidebar}[1]{%
  \insert\sideins{%
\begin{mdframed}[
  backgroundcolor=gray!30,
  skipabove=0pt,
  innertopmargin=0pt,
  userdefinedwidth=\sidebarwidth,
  frametitle=Tip:,
  frametitlebelowskip=0pt]
    \hsize\sidebarwidth
    \@parboxrestore
    \def\baselinestretch{\m@m@footnote@spacing}%
    \m@mwhich@margin{\m@msidebar@margin}%
    \sidebarform\sidebarfont
    \splittopskip=\ht\strutbox
    \splitmaxdepth=\dp\strutbox % doesn't do anything useful
    \allowbreak
    \prevdepth=\dp\strutbox    % supersedes a "top-strut"
    \vskip-\parskip
    \parskip=\smallskipamount% optional
    #1%
    \ifvmode\else
      \unskip\@finalstrut\strutbox
    \fi\par
    \ifdim\prevdepth>\dp\strutbox \prevdepth=\dp\strutbox \fi
    \ifdim\prevdepth>99\p@
      \nobreak
      \vskip-\prevdepth
      \allowbreak
      \vskip\dp\strutbox
    \fi
    \vskip\sidebarvsep\end{mdframed}}%
}
\makeatother

\begin{document}

\setsidebars{*}{3cm}{*}{*}{*}{*}
\lipsum[1-3]\sidebar{% 
Here's a really useful trick to comparing fractions. It only works for two 
fractions at one time, but you can use it a few times if you have a few fractions.
\par
You can cross-multiply and compare the products to see which fraction is larger.}

\end{document}