Your code use resembles what is defined by algpseudocode, not algcompatible. However, the same process holds here (see end of post).
I first found the definition of \ElsIf within algpseudocode.sty:
\algdef{C}[IF]{IF}{ElsIf}[1]{\algorithmicelse\ \algorithmicif\ #1\ \algorithmicthen}
Then, I replicated it for a new command \NoThenElseIf:
\newcommand{\algorithmicelsif}{\textbf{elsif}}
\algdef{C}[IF]{IF}{NoThenElseIf}[1]{\algorithmicelsif\ #1}

\documentclass{article}
\usepackage[noend]{algpseudocode}
% Original definition of \ElsIf from
% algpseudocode.sty (http://mirrors.ctan.org/macros/latex/contrib/algorithmicx/algpseudocode.sty)
% \algdef{C}[IF]{IF}{ElsIf}[1]{\algorithmicelse\ \algorithmicif\ #1\ \algorithmicthen}%
\newcommand{\algorithmicelsif}{\textbf{elsif}}
\algdef{C}[IF]{IF}{NoThenElseIf}[1]{\algorithmicelsif\ #1}
\begin{document}
\begin{algorithmic}
\If{c1}
\State True branch
\ElsIf{c2}
\State Else true branch
\NoThenElseIf{c3}
\State No \textbf{then} true branch
\EndIf
\end{algorithmic}
\end{document}
For algcompatible:
\algdef{C}[IF]{IF}{ELSIF}%
[2][default]{\algorithmicelse\ \algorithmicif\ #2\ \algorithmicthen\ALG@compatcomm{#1}}%
This can be replaced with
\makeatletter
\newcommand{\algorithmicelsif}{\textbf{elsif}}
\algdef{C}[IF]{IF}{NOTHENELSIF}%
[2][default]{\algorithmicelse\ \algorithmicif\ #2\ALG@compatcomm{#1}}%
\makeatother
Here's a complete minimal example:
\documentclass{article}
\usepackage[noend]{algcompatible}
% Original definition of \ELSIF from
% algcompatible.sty (http://mirrors.ctan.org/macros/latex/contrib/algorithmicx/algcompatible.sty)
% \algdef{C}[IF]{IF}{ELSIF}%
% [2][default]{\algorithmicelse\ \algorithmicif\ #2\ \algorithmicthen\ALG@compatcomm{#1}}%
\makeatletter
\newcommand{\algorithmicelsif}{\textbf{elsif}}
\algdef{C}[IF]{IF}{NOTHENELSIF}%
[2][default]{\algorithmicelse\ \algorithmicif\ #2\ALG@compatcomm{#1}}%
\makeatother
\begin{document}
\begin{algorithmic}
\IF{c1}
\STATE True branch
\ELSIF{c2}
\STATE Else true branch
\NOTHENELSIF{c3}
\STATE No \textbf{then} true branch
\ENDIF
\end{algorithmic}
\end{document}