As stated in the documentation, sys.last_type, sys.last_value and sys.last_traceback ... are set when an exception is not handled and the interpreter prints an error message and a stack traceback.

This is true whether the exception is a SyntaxError or some runtime error when a standard Python interpreter is used.

However, when the IDLE shell is used and a SyntaxError occurs, these are not set.  If it is not possible to change this behaviour in IDLE, perhaps the documentation of the sys module should be amended to include this information.

This is most likely an oversight.  It affects both normal and -n mode.  The three sys attributes can be written.  Given the error instance e caught in the IDLE process, we need to execute, *in the user namespace*,
  sys.last_type, sys.last_value, sys.traceback = type(e), e, None

The syntax error handle should call run_source (or maybe run_code).  I will make a PR if it seem to work.

If this can be implemented, then I believe that https://bugs.python.org/issue43476 could be closed as well.

The method to run just code in userspace, without involving the debugger or history list, is runcommand.  It needs a code string.  The hard part is the code needed to recreate the exception instance in the user space.  It will be different for SyntaxError versus the other error possible when compiling.  The SE details tuple has a mix of strs and ints.  I believe I know in principle how to do it.

#43476 is clearly an enhancement request.  It proposes to stash info in the GUI process in a way that lets it be into from the other process by non-standard means.

This is closer to a bugfix request in that successive lines have a different response in REPL and Shell.  Having sys attributes different in the two environments is not a bug in itself: for instance, sys.stdxxx *must* be different in the two environments because the standard streams *are* different.  Having sys.last_type, for instance, being either None or wrong (the last non-syntax error) instead of SyntaxError is something different.  I fix here will be harder, but I like it better.

If this is not fixed, perhaps it should be documented along with other IDLE differences in Running User code.