Divi Theme: Replace Top Header with a Header Sidebar Area

functions.php

// Register custom sidebar
function custom_register_sidebars() {
    register_sidebar( array(
        'name'          => 'Header',
        'id'            => 'header_sidebar',
        'description'   => 'Widgets in this area will be shown in the header.',
        'before_widget' => '<div id="%1$s" class="widget %2$s">',
        'after_widget'  => '</div>',
        'before_title'  => '<h2 class="widgettitle">',
        'after_title'   => '</h2>',
    ) );
}
add_action( 'widgets_init', 'custom_register_sidebars' );

// Hook into et_html_top_header filter
function custom_header_sidebar_content( $top_header ) {
    ob_start();

    // Check if Header sidebar has widgets
    if ( is_active_sidebar( 'header_sidebar' ) ) {
        echo '<div id="top-header">'; // Start the wrapper div
        dynamic_sidebar( 'header_sidebar' ); // Display widgets
        echo '</div>'; // Close the wrapper div
    } else {
        echo $top_header; // If no widgets, display original content
    }

    $content = ob_get_clean();

    return $content;
}
add_filter( 'et_html_top_header', 'custom_header_sidebar_content' );

Instructions:
Add some text in the Appearance -> Customizer -> Header & Navigation -> Header Elements -> Phone Number in order to trigger the top header showing. If a widget exists in the Header sidebar area it will replace the normal Divi Top Header content with the widget.

Pro Tip:
Use a plugin like https://wordpress.org/plugins/shortcodes-for-divi/ so you can build a header design in the Divi Library and then use the shortcode to add to the header widget.