On the client side you are, presumably, using a HTTP client (such as Spring's RestTemplate or Apache Commons' HttpClient). Something like this, perhaps:
ResponseEntity<String> response = restTemplate.getForEntity(url, String.class);
Given this code, the HTTP status and the response body are available as follows:
HttpStatus httpStatus = response.getStatusCode(); String responseBody = response.getBody();
So, whatever status code and response body you supplied via your RestControllerAdvice can be read out of the response.
A typical approach would be to assess whether the HttpStatus is a 200 and if it is not then fall into an exception handling block, using the response body to understand the nature of the exception.