On a WordPress multisite subsite track if the user is logged in on the main site and add log In or Log Out link to the primary menu

functions.php

<?php

/**
 * Check if the user is logged in to the main site.
 *
 * @return bool True if the user is logged in to the main site, false otherwise.
 */
function sjd_is_user_logged_in_main_site() {
	switch_to_blog( 1 );  // 1 usually represents the main site in a multisite setup.
	$logged_in = is_user_logged_in();
	restore_current_blog();  // Switches back to the original blog.
	return $logged_in;
}

/**
 * Add a custom login link to the primary menu.
 *
 * @param  string $items The HTML list content for the menu items.
 * @param  object $args  An object containing wp_nav_menu() arguments.
 * @return string The modified HTML list content for the menu items.
 */
function sjd_add_custom_login_link_to_nav( $items, $args ) {
	if ( 'primary' === $args->theme_location ) {  // Change 'primary' to your desired menu location if different.
		if ( sjd_is_user_logged_in_main_site() ) {
			// If the user is logged in, show a logout link.
			$logout_url   = wp_logout_url( get_site_url( 1 ) );  // Redirect to the main site after logout.
			$link_content = '<li><a href="' . esc_url( $logout_url ) . '">Log Out</a></li>';
		} else {
			// If the user is not logged in, show a login link.
			$main_site_login_url = get_site_url( 1, 'log-in', 'login' );
			$link_content        = '<li><a href="' . esc_url( $main_site_login_url ) . '">Log In</a></li>';
		}

		// Prepend the custom link to the beginning of the menu.
		$items = $link_content . $items;
	}

	return $items;
}
add_filter( 'wp_nav_menu_items', 'sjd_add_custom_login_link_to_nav', 10, 2 );