I had an attempt to implement this a while ago, but then abandoned it (and instead tried to find shortest URLs, or in the worst case use \footenotesize or something like that in bibliography, such that the link fits on one line).

However, I gave it another try, any following seems promising. There are still some spacing issues, since the \refbox is around the invisible \rule with zero thickness (change \rule[-0.2ex]{\ulwidth}{0.0pt} to see it), not actually around the text. Perhaps it will be enough for you (after you fix the spacing in case it bothers you).

\documentclass[a4paper,12pt]{article}

\usepackage{lipsum}
\usepackage{multicol}

\AddToHook{package/soulpos/before}{\RequirePackage{soul}}
\usepackage{soulpos}

\usepackage{tcolorbox}
\tcbuselibrary{skins, breakable, theorems}

\tcbset{
    link/.style={
        nobeforeafter, box align=base, boxsep=0em, before upper={\vphantom{]}},
        left=0.15em, right=0.15em, top=0.25ex, bottom=0.05ex,
        arc=0.1em, rounded corners=all, boxrule=0.3mm,
        grow sidewards by=0.05em, enlarge bottom by=-0.05ex, enlarge top by=-0.4ex,
        grow sidewards by=0.15em+0.3mm,
    }
}
\tcbset{
    refbox/.style ={colframe=black!30, colback=black!3},
}
\tcbset{
    continuestart/.style={leftrule=0mm, outer arc=0.1em+0.3mm, rounded corners=east},
    continueend/.style={rightrule=0mm, outer arc=0.1em+0.3mm, rounded corners=west},
}
\NewTCBox{\refbox}{!O{}}{link,refbox,#1}

\makeatletter
\ulposdef{\myul}[gap=0.4em]{%
    \ifulstarttype{0}{\def\box@start{}}{\def\box@start{continuestart}}%
    \ifulendtype{0}{\def\box@end{}}{\def\box@end{continueend}}%
    \refbox[\box@start,\box@end]{\rule[-0.2ex]{\ulwidth}{0.0pt}}%
}
\makeatother

\begin{document}

Lorem ipsum dolor sit amet, \myul{consectetur adipiscing} elit.
Nunc facilisis vitae massa nec cursus.
Fusce faucibus gravida dolor.
Cras egestas eget nulla ut consequat.
Nulla eu mi at mauris feugiat ornare.
Etiam a efficitur sem, et scelerisque massa.
Sed ipsum justo, posuere vitae leo vitae, fermentum dictum lacus.
Morbi et ipsum vel neque congue elementum.
In dolor orci, hendrerit vitae sagittis sit amet, pellentesque a magna.

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Nunc facilisis vitae massa nec cursus.
Fusce faucibus gravida dolor.
Cras egestas \myul{eget nulla ut consequat.
    Nulla eu mi at mauris feugiat ornare.
    Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed} diam voluptua.
At vero eos et accusam et justo duo dolores et ea rebum.
Stet clita kasd, no sea est Lorem ipsum dolor sit amet.
aksdh salkdj salkdjh.
\myul{Etiam a efficitur sem, et scelerisque massa.
    Sed ipsum justo, posuere vitae leo vitae, fermentum dictum lacus.}
Morbi et ipsum vel neque congue elementum.
In dolor orci, hendrerit vitae sagittis sit amet, pellentesque a magna.

\begin{multicols}{2}
    Lorem ipsum dolor sit amet, consectetur \myul{adipiscing} elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
    \newcolumn

    Lorem ipsum dolor sit amet, consectetur \myul{adipiscing elit, sed do eiusmod tempor} incididunt ut labore et dolore magna aliqua.
\end{multicols}

\end{document}

Automatic line break for long URLs #20

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