It took me long 2 hours to figure out that simple code, but hopes it will help someone in developing wordpress theme :-) Result is style link with rel='alternate stylesheet' Code below:

function theme_styles(){
    global $wp_styles;  //key to the case
    // NORMAL STYLE
    wp_enqueue_style('main-css', 'main.css');

    // Alternate style
    wp_enqueue_style('alt-css', 'alt.css');
    $wp_styles->add_data('alt-css','alt', true); //same handel as above, extra data 'alt' and true ;-)
}
add_action( 'wp_print_styles', 'theme_styles' );

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