Path-based rules don't apply

Correct and not fixable. This is a trade off for having domain-specific paths.

There is a potential feature request here for setting Context conditions properly, but the core path-lookups are hard-wired and cannot, AFAIK, be overridden.

It might be helpful to anyone who also got this issue.

The idea is to write a custom context path plugin

Here is the code

Define plugin in the mymodule.module

/**
 * Implements hook_context_plugin().
 */
function mymodule_context_plugins() {
    $plugins = array();
    $plugins['context_mymodule_path'] = array(
        'handler' => array(
            'path' => drupal_get_path('module', 'mymodule') .'/plugins',
            'file' => 'context_mymodule_path.inc',
            'class' => 'context_mymodule_path',
            'parent' => 'context_condition',
        ),
    );

    return $plugins;
}

/**
 * Implements hook_context_registry().
 */
function mymodule_context_registry() {
    $registry = array();

    $registry['conditions'] = array(
        'mymodule_path' => array(
            'title' => t('Compatible Path'),
            'description' => t('Make this path condition compatible with domain_path'),
            'plugin' => 'context_mymodule_path',
        ),
    );

    return $registry;
}

Save this code to mymodule/plugins/context_mymodule_path.inc

class context_mymodule_path extends context_condition_path {

    /**
     * Execute.
     */
    function execute() {
        if ($this->condition_used()) {
            // Include both the path alias and normal path for matching.
            $current_path = array($_GET['q']);

            $path_alias = drupal_get_path_alias($_GET['q']);

            if ($path_alias == $_GET['q'] && module_exists('domain_path')) {
                $path_alias = domain_path_lookup_path('alias', $_GET['q']);
            }

            // $path_alias can be false.
            if ($path_alias && $path_alias != $_GET['q']) {
                $current_path[] = $path_alias;
            }

            foreach ($this->get_contexts() as $context) {
                $paths = $this->fetch_from_context($context, 'values');
                if ($this->match($current_path, $paths, TRUE)) {
                    $this->condition_met($context);
                }
            }
        }
    }
}

And set the trigger in the mymodule.module

/**
 * Implementation of hook_init().
 */
function mymodule_init() {
    if ($plugin = context_get_plugin('condition', 'mymodule_path')) {
        $plugin->execute();
    }
}