I've found a solution.
You can get response headers using this after you've made the call:
((BindingProvider)port).getResponseContext().get(MessageContext.HTTP_RESPONSE_HEADERS);
Find the Set-Cookie
header and store its value.
Then before your next request (in any webservice) you can set the Cookie header:
((BindingProvider)port).getRequestContext().put(
MessageContext.HTTP_REQUEST_HEADERS,
Collections.singletonMap("Cookie", Collections.singletonList(cookieValue)
)
);