libghostty: C api to stream formatter output through a GhosttyWriter

Add `ghostty_formatter_format` which uses a streaming GhosttyWriter
type to write. Update the example to show this.

include/ghostty/vt/formatter.h

@@ -11,6 +11,7 @@
 #include <stddef.h>
 #include <stdint.h>
 #include <ghostty/vt/allocator.h>
+#include <ghostty/vt/io.h>
 #include <ghostty/vt/selection.h>
 #include <ghostty/vt/types.h>
 #include <ghostty/vt/terminal.h>
@@ -137,6 +138,30 @@ GHOSTTY_API GhosttyResult ghostty_formatter_terminal_new(
     GhosttyTerminal terminal,
     GhosttyFormatterTerminalOptions options);
 
+/**
+ * Run the formatter and stream output to a writer.
+ *
+ * Each call formats the current terminal state and invokes the writer
+ * synchronously as output becomes available. The callback may be called more
+ * than once and must not call formatter or terminal APIs using the same
+ * formatter or its terminal.
+ *
+ * If an error occurs, the writer may already contain a partial formatted
+ * output. The operation cannot be resumed from that partial output. This
+ * function does not flush or make the caller's destination durable.
+ *
+ * @param formatter The formatter handle (must not be NULL)
+ * @param writer Destination writer whose write callback must not be NULL
+ * @return GHOSTTY_SUCCESS on success, GHOSTTY_IO_ERROR if the writer rejects
+ *         output, GHOSTTY_LIMIT_EXCEEDED if output accounting overflows, or
+ *         GHOSTTY_INVALID_VALUE if an argument is invalid
+ *
+ * @ingroup formatter
+ */
+GHOSTTY_API GhosttyResult ghostty_formatter_format(
+    GhosttyFormatter formatter,
+    GhosttyWriter writer);
+
 /**
  * Run the formatter and produce output into the caller-provided buffer.
  *