Yes you can have single login for both Joomla project and you core php project.
This can be done by creating a user plugin that is called on login and if successful the authenticated account details can be passed to other project via http request, rest or via a database stored variables.
Heare is demostration using a http return link;
<?php
class plgUserPLUGINNAME extends JPlugin {
onUserLogin($user, $option){
$application = JFactory::getApplication();
if($user['id']){
$return = JRequest::getVar('return');
$application->setRedirect($return);
return true;
}else{
return false;
}
}
onUserLogout($credentials){
$application = JFactory::getApplication();
$return = JRequest::getVar('return');
$application->setRedirect($return);
return true;
}
?>
Note You will access the login within joomla and redirect will happen based on return url.
To call the page from core php project
<?php
$token = rand(0,100);
$_SESSION['login_token'] = $token;
$return = 'www.corephpproject.com/loggedin.php&token=$token';
?>
<a href="www.yourjoomlasite.com?option=com_users&view=login&return="<?php echo $return; ?>>Login</a>
To call from joomla you will us joomla menu links and modules.