next_post_link and previous_post_link in custom post type single page, in same non-custom category

First please check your “custom_post_type” function that you Register Custom Post Type is everything correct like 'taxonomies'=> array( 'category') & 'has_archive' => true, ! If so then you must be put wrong way to display it.

About next_post_link and previous_post_link Please try with like this inside your single-realisation.php file

$prev_post = get_previous_post(); $next_post = get_next_post();    
 if(!empty($prev_post)):
        echo '<a href="'.get_the_permalink($prev_post->ID).'">'.__('Previous','text-domain').'</a>';
  endif;

if(!empty($next_post)):
     echo '<a href="'.get_the_permalink($next_post->ID).'" >'.__('Next','text-domain').'</a>';
endif;

I believe you will get your next & Previous post link form your custom post type also;

Thanks
Msua