I figured something out based on what I found at this post.

Basically I need a plugin that listens for when the REST response is about to go out. The plugin code would be similar to the following:

function ag_filter_post_json($response, $post, $context) {
    $tags = wp_get_post_tags($post->ID);
    $response->data['tag_names'] = [];

    foreach ($tags as $tag) {
        $response->data['tag_names'][] = $tag->name;
    }

    return $response;
}

add_filter( 'rest_prepare_post', 'ag_filter_post_json', 10, 3 );

It adds the tag names as a new property called tag_names. The rest of the heavy lifting is done by the wp_get_post_tags function.

I figured something out based on what I found at this post.

Basically I need a plugin that listens for when the REST response is about to go out. The plugin code would be similar to the following:

function ag_filter_post_json($response, $post, $context) {
    $tags = wp_get_post_tags($post->ID);
    $response->data['tag_names'] = [];

    foreach ($tags as $tag) {
        $response->data['tag_names'][] = $tag->name;
    }

    return $response;
}

add_filter( 'rest_prepare_post', 'ag_filter_post_json', 10, 3 );

It adds the tag names as a new property called tag_names. The rest of the heavy lifting is done by the wp_get_post_tags function.