I'm with bhofcc, as far as this being an irradiated bottle.
It is not scarce. I have dug them. The club sauce bottles just do not get any respect, in praise, or dollars.
"A & P ……………Uncertain. Reported on the base of a sauce bottle (a Worcestershire Sauce “lookalike” brand) dating circa 1880s. In this case the mark is probably that of the food product maker or distributor, and not that of the glass maker."
GLASS MANUFACTURERS' MARKS ON BOTTLES & OTHER GLASSWARE
" Worcestershire Sauce - The Rivals.
Lea and Perrins has not been the only Worcestershire sauce to be made. Its early success encouraged other firms to copy the recipe in competition. In 1906, Lea and Perrins took legal action against a Birmingham sauce manufacturer called Holbrooks, to try and restrict the use of the name "Worcestershire Sauce". The court decided that the name could be used by anyone, but only Lea and Perrins had the right to call theirs 'Original and Genuine'
Sauces were particularly popular during the 19th and 20th centuries as they gave flavour to otherwise plain food and helped tenderise tough cuts of meat. They were also useful for disguising the flavour of foods that were past their best."
Worcester City Art Gallery & Museum - Spirit of Enterprise Exhibition - Wine & Sauce Making

"At one time there were 6 separate sauce factories in Worcester, England, each producing the thin, market-leading Worcester sauce.
Lea & Perrins
Four Flags
Mellor & Co.
Courtney’s.
Waldren and Co.
Stretton and Co."
Early Bird Worcester Sauce