When the King is dying, the paper is not under his pillow in one scene and then magically it is there when he takes it out to be read.
As the various Consulates are being established, there are several errors involving the national flags; the British Union Jack is shown upside down, the French flag is replaced by that of The Netherlands, and the US flag is shown with 48 stars (correct for 1946, but in 1870 there were only 37 states).
In order to remind the King about his promise to give Anna a house of her own (to which the King continually refuses to honour) she leaves many reminders about his broken promise around the palace, in particular an English china cottage which has "made in England" on the underside and the inscription over the door of the cottage "Honor Thy Promise". In 1868 England (as today), 'honour' was not spelt "honor". Minor error, but a spelling mistake non-the-less.
A calendar with the months of May and June is shown to demonstrate how long Anna has waited for her audience with the King. June 1st to June 9th is crossed out and half of May is crossed out as well. But it is May 1st to 16th that is crossed out and May 17th to 31st is untouched. It should have been just the opposite.
The banquet scene includes a performance of Siamese theatrical dance. However, the music which accompanies this performance is unmistakably Balinese gamelan, not Siamese. Furthermore, it's "gong kebyar" style, which hadn't been invented yet in the 1860s.
Anna wears her hair down, in the style of 1946. Not only is this an anachronism, but at the time it would have been considered vulgar and bizarre for a married woman or a woman of Anna's age (and she is both) not to wear her hair up.