threefold repetition or not?

If play continues Kf8 Bg6 Bh3 Rf7+ Ke8 Bh5 Bg2+ Rb7+ then the same position is reached. If this sequence is played once more, and Black claims a threefold repetition, how should an arbiter rule?

The relevant FIDE rule "Positions as in (a) and (b) are considered the same, if the same player has the move, pieces of the same kind and colour occupy the same squares, and the possible moves of all the pieces of both players are the same.
Positions are not the same if a pawn that could have been captured en passant can no longer be captured in this manner. When a king or a rook is forced to move, it will lose its castling rights, if any, only after it is moved"

is unclear. Black may have castling rights in the initial position, but could not possibly exercise them in any continuation.

Actually black could play g6 and white could resign.  Regardless, the rules are pretty clear.

This is an interesting position.

I might be wrong, but I don't believe it's a repetition of position yet... for the point you indicated. 

In the first position, the diagram you posted, Black hadn't castled yet so he still had the right to castle.  He COULD capture the bishop giving check with his rook or he could move his king.  AFTER either move, of course, he's now lost the right to castle, but this right hasn't yet occured until AFTER the move is played. 

So from the given position, the castling rights are still "yes."  From a computer programmer's point of view, the "can I still castle kingside" flag/switch would be still be set to "yes."  This flag/switch doesn't get set to "no" until AFTER Black moves.

But then as you indicate, when the position occurs again, we're really seeing this same position for the FIRST time, not the second time.  From this position, the castling rights are now "no."

An arbiter who is not as smart as me might see it differently.  

I think the key word is "rights."  Black HAS castling rights in the first position, even though, again as you point out, he can't exercise those rights for this move, since he's in check.

And, again, not to belabour the point, Black plays g6 and we will not reach this position again.

I remember Karpov once was in such a situation. He gave a check in a position where his opponents King hadn't moved yet. It still had the right to castle (not while in check ofcourse) but had to move. Karpov then gave check again (same piece) and then he repeated these moves 3-times. His opponent then claimed a draw but the arbiter corrected him. In the final position the King no longer has the right to castle , but in the initial position he did ! So the positions aren't the same. Karpov then played another move and went on to win the game.

You're missing the whole point.  We're not talking about what the best move might or might not be.  I don't understand why you keep bringing this up.

I believe I read something by Geurt Gijssen on chesscafe.com where he said castling rights exist in regard to three-fold repetition if the king and rook haven't moved, regardless of whether it is legal in that game (i.e., even if the only legal moves are by the king or rook). Similarly, en passant rights exist in regard to this if the pawn has just advanced two squares, even if the en passant capture is illegal (i.e., a pinned pawn).

Well, Ed, the contention was that the position could be reached again - it could not.  Why an arbiter who "was not as smart as you" would fail to grasp a rule of the game, I do not know.  The rules are clear, it is not the same position unless it really is the same position.  Game states apply.  To actually claim I am missing the point is to miss the truth.  The OP even quoted the exact rule and unless you want to delete part of that rule, I see no confusion.

I think a good rule of thumb would be to compare the FEN string, only if that matches exactly is the position the same.

EDIT: disregarding of course the halfmove and fullmove counters

How long is a piece of F'n string?

I think what Here_Is_Plenty is saying, is that until black plays ...Kf8, the position hasn't completely repeated three times. If black chooses to play ...g6, then there is no threefold repetition. The reason being, is that the starting position in the diagram is different because black has the option to castle in this position, and therefore the full repetition begins after black has played ...Kf8.

I think the formulation of 9.2 could be improved a lot. "Possible moves" is undefined for a player when its not her turn. "Can be captured in this manner" doesn't say whether pins matter. "king or a rook is forced to move" ignores unforced king/rook moves". Worst of all, castling rights are not explicitly mentioned in the main defining sentence.

A much clearer formulation would be:

"Positions as in (a) and (b) are considered the same, if the same player has the move, pieces of the same kind and colour occupy the same squares, the set of en-passant pawns is the same, and castling rights are the same.

An en-passant pawn is one that moved 2 spaces on the last turn and is adjacent to an opponent pawn on its rank. A castling right exists for a king and same colored rook if neither has yet moved."

If the right to castle is "immaterial and irrelevant" in looking at threefold repetitions, then why does FIDE specifically refer to them in Article 9.2 ?...

"Positions are not the same if a pawn that could have been captured en passant can no longer be captured in this manner. When a king or a rook is forced to move, it will lose its castling rights, if any, only after it is moved."

Of course, the bit in red is never possible :)

There is such a thing as the empty set...

Yeres do you understand that words in everyday language may be used in technical applications to have a slightly less obvious meaning?  Terms which we would define one way in straightforward language sometimes have definitions, qualified or demonstrated in Articles.... This does not just apply to chess.

I don't think so. If you read the rules more closely, it probably definies "identical position" somewhere as considering the right to castle and capture via en passant.