Dashboard and Site Language only partially translated (in own theme)

Finally I found the issue!

The following filter was causing the theme to only translate partially:

function howdy_message($translated_text, $text, $domain) {
    $new_message = str_replace('Howdy', 'Aloha', $text);
    return $new_message;
}
add_filter('gettext', 'howdy_message', 10, 3);

I replaced it with the following that now works:

function howdy_message($translated, $text, $domain) {
    if (false !== strpos($text, 'Howdy'))
        return str_replace('Howdy', 'Aloha', $text);

    return $translated;
}
add_filter('gettext', 'howdy_message', 10, 3);

Why, I don’t know, but by stripping down my function.php bit by bit I managed to find the function that was causing the issue.