The Gemini 10 embroidered patch is clearly visible on the spacesuits of Collins and Young during the suiting-up for the mission itself as shown here in S66-42424.
The actual mission patch worn by John Young on his space suit was sold along with the US flag patch and name tag patch as a lot in the Spring 2009 Heritage space auction.
John Bisney's research for his forthcoming book on space patches has revealed that this patch was manufactured by Texas Art Embroidery.
Further research by Kenny Suit has shown that just 110 examples were supplied to NASA.
This variant of the Crew Patch has blue background thread which has horizontal rows about 50% wider than those of the standard crew version. This is particularly noticeable on the reverse of the patch.
In general, the black thread used around the "X" and on the spacecraft orbits is noticeably thinner in this version than the crew version. This makes the Gemini capsule skinny and the Agena particularly slim. There is also a clear gap behind the Agena in this example that is not present in any examples of the other variant that I've seen.
One point on the crew version where the black thread is not thick is at the upper left side of the lower yellow star, where the black becomes a single line of thread. This feature is consistent on all examples of the crew patch I've seen to date. On the variant the black thread is of more or less constant thickness all the way round the star.
Also noteworthy is that the extreme edges of the crew versions that I've seen all have trimmed traces of the pale blue backing cloth visible all around whilst this version has none. The edges appear black all round indicating that this version was apparently embroidered on black cheesecloth.
These differences are more than just natural variations during a production run, they indicate a separate production using a different pattern. Because the design is so similar to the crew version it seems likely that it was produced by the same company producing a separate run of the patch from a new pattern based on the same artwork. Whether it was produced at the same time as the other variant or sometime afterwards is impossible to say.
A high quality modern replica of the Gemini 10 crew patch, produced by AB Emblem in 2010 in collaboration with Gene Dorr.
The standard AB Emblem souvenir version of the patch has the "X" entirely enclosed in the circular design and adds the crew names.
A replica patch produced in the late 80s by Eagle One Aerospace. Not as convincing as some of their other replicas. In particular the color is way off and the extremities of the "X" are rounded on this version but squared-off on the real thing.
[1] Examples of Crew Patch sales :
Heritage Auctions Sep 21, 2007 lot 25083 sold at $4182.50 - unflown
Novaspace Auction Jan 26 2008 lot 407 sold at $360 - unflown from MC
Lunar Legacies Space Auction Mar 14, 2009 lot no. 117 sold at $1,338 - unflown
Heritage Auctions Apr 1, 2009 lot 41103 sold at $23,900 - FLOWN from Young's spacesuit, with other items
Heritage Auctions Apr 1, 2009 lot 41102 - unflown patch from JY with flown items
Heritage Auctions Apr 1, 2009 lot 41101 sold at $2,629 - FLOWN from JY
Heritage Auctions Oct 8, 2009 lot 41033 sold at $1,792.50 - FLOWN from JY
Heritage Auctions Apr 21, 2010 lot 41033 UNSOLD at $1,195 - FLOWN from JY
Heritage auction lot 41044 Jun 3 2011 sold at $1673 - FLOWN from JY
Heritage auction lot 40084 Nov 3 2013 sold at $2250 - FLOWN from JY
Heritage auction lot 50668 May 10, 2019 sold at $5,250 - FLOWN from JY
Sothebys Space Auction lot 54 Jul 20, 2019 sold at $1,250 FLOWN from JY
Heritage Auction lot 50450 Nov 14, 2019 sold at $2,000 - FLOWN from JY
RRAuction Oct 15, 2020, Lot 4050 at $1,250 - FLOWN from JY
Heritage Auction Dec 6, 2024 Lot 50021 sold at $2,375 [FLOWN by MC]
[2] Examples of Crew Patch Variant sales :
eBay item no. 360228196627 Jan 31. 2010, sold at $551
RR Auction lot 158 Apr 22, 2014 sold at $857.50 (unflown, mounted with Young ISP)