Here's a different approach to the problem which requires less manual intervention, and has an intuitive syntax. It also allows you to use it as an \edge style on an existing branch of the tree without specifying any nodes at all.
\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-qtree}
\usepackage{tipa}
\usetikzlibrary{decorations,decorations.text}
\tikzset{delink/.style={
decorate,
postaction={decorate,
decoration={text along path,
text align=center,
text={|\tiny|=},
transform={rotate=90}}}}}
\begin{document}
\begin{tikzpicture}[baseline]
\tikzset{frontier/.style={distance from root=85pt}}
\Tree
[.$\omega$ \edge[delink];
[.$\sigma$ \textipa{P} [.$\mu$ e ] [.$\mu$ \node(a){t}; ] ]
[.\node(x){$\sigma$}; [.\node(y){$\mu$}; e ] ]
[.\node(b){$\sigma$}; \edge[delink]; n [.$\mu$ \node(c){e}; ] [.\node(d){$\mu$}; ] ]
[.$\sigma$ h [.$\mu$ \node(f){u}; ] [.\node(e){$\mu$}; ] ]
]
\draw[delink] (d.south) -- (c.north);
\draw[delink] (e.south) -- (f.north);
\draw[delink] (x.south) -- (a.north);
\end{tikzpicture}
\end{document}

A forest solution
The forest package provides a simpler way to add named nodes to trees, so personally I would use forest to do these kinds of trees. The same delink style can be used with it too. Here's the same code using forest. I've added code to automatically format the segmental tier using tipa (so you don't need to wrap them in \textipa) and then each segment will be in the same font. For this to work, if a mora node has direct no segmental dependent (i.e., it is linked to a segment that is already dominated by a mora node), you need to explicitly state that the node is on the mora tier.
\documentclass{article}
\usepackage{tikz}
\usepackage[linguistics]{forest}
\usepackage{tipa}
\usetikzlibrary{decorations,decorations.text}
\tikzset{delink/.style={
decorate,
postaction={decorate,
decoration={text along path,
text align=center,
text={|\tiny|=},
transform={rotate=90}}}}}
\newcommand{\syl}{$\sigma$}
\newcommand{\mor}{$\mu$}
\newcommand{\word}{$\omega$}
\forestset{prosodic tree/.style=
{for tree={if n children=0{tier=seg,font=\tipaencoding}{},inner sep=0pt,s sep=1em}}}
\begin{document}
\begin{forest}prosodic tree
[\word
[\syl,edge=delink [P] [\mor [e] ] [\mor [t,name=a ] ] ]
[\syl,name=x [\mor,name=y [e] ] ]
[\syl,name=b [n] [\mor [e,name=c ] ] [\mor,name=d,tier=mora ] ]
[\syl [h] [\mor [u,name=f ] ] [\mor,name=e,tier=mora ] ]
]
{\draw[delink] (d.south) -- (c.north);
\draw[delink] (e.south) -- (f.north);
\draw[delink] (x.south) -- (a.north);}
\end{forest}
\end{document}

forestpackage, I defined a new edge style like so:strike/.style={ edge label={ node[midway, sloped, rotate=90] {=} } }– hftf Jul 17 '17 at 08:14