Re Tesco Groceries website...

Several days ago I upgraded to Mavericks on my MacBook, and have had no problems with accessing websites on Safari until starting an order today on Tesco Groceries' website (which had presented no problems before the upgrade). Signing-in appeared to be no problem, and the website displayed my name as usual. Searching for goods also was not problematic, but when I tried to add an item to my basket, the message

We need to check if your store has this product. Simply sign in or register on the next page.

appeared (this would have made sense if I had not yet signed in); I pressed "Continue", and the sign-in page again appeared before me; when I signed in, the item appeared successfully in my basket. I continued to build up my order in this wonky way, being prompted to sign in each time I tried to add an item to my 'basket'.

When I pressed the key to check-out, again the sign-in page anomalously appeared before me; after I duly signed in again, the usual options for delivery/collection appeared; selecting delivery, again I was anomalously prompted to sign in, which I did; but unfortunately my 'luck' ran out there as when I selected a date and time, and was again anomalously prompted to sign in and did so, my choice of date and time was not registered...I tried several times to select date and time, and each time was prompted to sign in again, but my chosen date and time was not saved.

At that point I contacted Tesco and spoke to someone in technical support, who advised me to sign out fully, leave the page, clear out Tesco cookies and sign in again. My previous 'basket' of selected items reappeared, but the same problems with anomalous prompts to sign in persisted. I also tried removing an item from my 'basket', which again initiated the sign-in screen, but after I signed in again the item was still in the 'basket' (i.e the attempt to remove it did not work). The technical assistant suggested trying with Firefox to see if the problem repeated itself in a different browser. I have since tried with Firefox and was successful, meeting none of the problems stated above.

Have others had this problem with Tesco Groceries and the Mavericks update? (I have noted that Mavericks users had a problem with accessing Tesco Bank last year.) Have others any solutions (apart from sticking with Firefox for Tesco Groceries, at least for now)?