可以通过在API控制器中使用HttpResponseMessage来同时返回附件和JSON数据。首先，可以将附件转换为一个HttpResponseMessage对象，然后将JSON数据转换为另一个HttpResponseMessage对象，并将两者合并为一个HttpResponseMessage对象，最后返回该对象。下面是一个示例代码：

public HttpResponseMessage Get()
{
    // 获取要返回的附件
    byte[] fileBytes = ReadFile();

    // 创建附件的HttpResponseMessage对象
    HttpResponseMessage attachmentResponse = new HttpResponseMessage(HttpStatusCode.OK)
    {
        Content = new ByteArrayContent(fileBytes)
    };
    attachmentResponse.Content.Headers.ContentDisposition = new ContentDispositionHeaderValue("attachment")
    {
        FileName = "file.pdf"
    };
    attachmentResponse.Content.Headers.ContentType = new MediaTypeHeaderValue("application/pdf");

    // 创建JSON数据的HttpResponseMessage对象
    MyObject myObject = new MyObject();
    HttpResponseMessage jsonResponse = Request.CreateResponse(HttpStatusCode.OK, myObject, Configuration.Formatters.JsonFormatter);

    // 将两个HttpResponseMessage对象合并为一个
    HttpResponseMessage finalResponse = new HttpResponseMessage(HttpStatusCode.OK);
    finalResponse.Content = new MultipartContent("mixed");
    finalResponse.Content.Add(new HttpMessageContent(attachmentResponse));
    finalResponse.Content.Add(new HttpMessageContent(jsonResponse));

    return finalResponse;
}