It would be better to send back a result form the ajax request and show/don't show the alert in the success callback:

$.ajax({
   type: "POST",
   url: "send.php",
   data: "name=" + called,,
   success: function(data) {    
      if ( data == "show" ) {
         // something happens
         alert("Correct name");
      } else {
         // alert box wasn't shown, something else happens.  
      }  
   }               
});

And on your server:

if ( $_POST['name'] == 'john' ) {
   echo "show";
}

You could use json_encode() php function to return data from php.

This will be a better approach :

PHP :

if (!isset($_POST['name'] || empty($_POST['name']) {
    die(json_encode(array('return' => false, 'error' => "Name was not set or is empty")));
} elseif ($_POST['name'] == "John") {
    die(json_encode(array('return' => true)));
} else {
    die(json_encode(array('return' => false, 'error' => "Name is different than John.")));
}

At this point, you will be allowed to check the returned values from JS and decide if you need to display the success alert or send an error message to the console (or do what ever you want...).

JS :

var called = $("#called").val();

$.ajax({
   type: "POST",
   url: "send.php",
   dataType: "JSON", // set the type of returned data to json
   data: {name: called}, // use more readable syntaxe
   success: function(data) {
      if (data.return) { // access the json data object 
         alert("Congrats ! Your name is John !");
      } else {
         console.log("Sorry, something went wrong : " + data.error);  
      }  
   }               
});

So, json_encode() allows to return easy accessible object from JS and will also allows you to set and display error messages easily in case the return is false.

Hope it helps !

PHP does not know if an alert has been shown, because in javascript the alert() function has no return value and no events which you could use to send an ajax request a click confirmation to the server.

One solution is to use a confirm() command inside the success event of your $.ajax(), which sends anothe ajax request if the user clicked "ok" or "cancel".

Something like this

var called = $("#called").val();

$.ajax({
   type: "POST",
   url: "send.php",
   data: "name=" + called,
   success: function(data) {

        if (data == "show") {

            var clicked = confirm("Correct name");

            if (clicked == true || clicked == false) {
                $.ajax({
                    url: "send.php?clicked=1",
                });
            }  
        }
        else {
             // Whatever to do than...
        } 
    }
});