I'm not sure how this "quartering" works it the sentence

I’d like to ask on the sentence from The Noble Bachelor by Conan Doyle.

.. will not prevent our children from being some day citizens of the
same world-wide country under a flag which shall be a quartering of
the Union Jack with the Stars and Stripes.

I’d like to know what kind of flag the speaker (Holmes) is referring to? Can you picture it?
I pictured one with quarter-sized Stars and Stripes being on the upper left side of the Union Jack background, as the same way Australian or New Zealand’s flag has the Union Jack design on theirs. With this design, Holmes is implying the UK will be colonized by the US in the near future?
Or is it the other way around? Quarter-sized Union Jack being on the Stars and Stripes?
Or is this sentence simply refering the Union Jack with the Stars and Stripes share their pattern in one flag? No master-servant implication at all? I don’t know.
Could someone tell me what kind of flag will be completed by this description please? Thanks.