I have submitted a ticket to the JOSM developers. The ticket contains a fully 
worked-out patch to upgrade the EPSG:27700 projection from the Helmert 
transformation to the look-up table transformation.

I'm afraid this is another somewhat long and technical message. But it does 
contain an explanation of why the Land Registry wms is actually sort-of correct.

With my modified version of JOSM, .mif files are now transformed using the 
look-up table. This is with the hack I described before, where no projection is 
defined in the file, and you choose EPSG:27700 when the opendata plugin asks 
you. But .shp files which declare British National Grid are still transformed 
with the Helmert transformation. This must be down to the opendata plugin, 
because JOSM itself, as modified, does not know about the Helmert 
transformation any more. (I have put in a ticket for that, too, but without a 
patch.)

I now understand better, the information in the EPSG registry. If you go to the 
page for projection 27700, and open the panel for the OSGB36 datum, it is not 
entirely clear, as I described before. But if you look at the page for the 
OSGB36 datum (transformation 7953), it spells out that this transformation uses 
the OSTN15 look-up table. However, transformation 7953 isn't referenced 
anywhere in the page for projection 27700.

The more I look at this projection stuff, the worse it gets.

The Land Registry only covers England and Wales. The Scottish counterpart is 
Registers of Scotland. I've also had a look at their opendata. The URL is a bit 
hard to find so I give it here https://ros.locationcentre.co.uk/inspire/ They 
too have a wms, which is here 
http://ros.datafeed.locationcentre.co.uk/geoserver/wms They offer .shp files in 
British National Grid and ETRS. I've already mentioned the issues raised by 
.shp files. Rob was hoping to compare the BNG and ETRS files, but that's not 
possible because Registers of Scotland have made a mistake in preparing the 
ETRS files. The BNG and ETRS files are identical. In other words, the ETRS file 
contains BNG coordinates in metres. I am a bit surprised that I might be the 
first to spot this and report it. (With shapefiles, the projection is defined 
in an accompanying .prj file. The .prj file uses a version of 'well-known text' 
which does not contain EPSG numbers. This will be part of the explanation for 
the issue with the opendata plugin. If the .prj file is missing, the only 
option with the opendata plugin is EPSG:4326 (WGS84), so removing the .prj file 
does not provide a workaround.)

The Registers of Scotland wms is misaligned, just like the Land Registry one. 
Rob's examples align with the two wms. *But* I always launch JOSM from the 
command line. And I spotted an info message on the command line saying 
'reprojecting from EPSG:27700'. So, a further discovery. Some background - We 
are familiar with online maps from various sources. Most of them use tms 
protocol with Web Mercator projection (EPSG:3857). Wms is different. The server 
offers the client a list of projections which it can deliver, and the client 
chooses which one to have. Suppose JOSM is set to EPSG:3857. The Land Registry 
wms does not offer EPSG:3857 so JOSM chooses the first projection which it 
understands, from the server's list - EPSG:27700. Then JOSM reprojects it, so 
the wms is misaligned because *JOSM* is using Helmert. And the Land Registry 
was right all along! The Registers of Scotland wms does offer EPSG:3857, so 
that is what JOSM chooses. And the wms is misaligned because the *wms server* 
is using Helmert. The wms also offers EPSG:27700. So if you set JOSM to 
EPSG:27700; and delete and re-add the wms layer so it is redownloaded in 
EPSG:27700; then the wms is misaligned because *JOSM* is using Helmert. [I mean 
delete the layer, not delete the entry in the imagery list.]

With my modified version of JOSM, both wms are correctly aligned (provided, in 
the case of Registers of Scotland, that you set JOSM to EPSG:27700 before 
adding the wms layer). The alignment of Rob's examples does not change with my 
modified version of JOSM, so Rob's examples no longer align with the two wms.

I should add that the other projections offered by the two wms servers, in as 
far as I have been able to test them, are all misaligned. In other words, both 
servers are using Helmert. I tested with JOSM and QGIS.

I've had another look at proj6 and proj7 and they do in fact use the look-up 
table for EPSG:27700. One piece I found online suggests they fall back to the 
Helmert transformation if the look-up table file is not available. This 
explains how come QGIS is using the look-up table transformation for EPSG:27700.

I've looked further into the issue of simplification of polygons (dropping of 
nodes). GDAL and ogr2ogr behave the same as QGIS. ogr2ogr has a simplify option 
but that only increases the amount of simplification: it won't reduce it below 
the preset minimum. QGIS uses GDAL, so it looks like the simplification is 
hard-wired into the output side of GDAL. The opendata plugin simplifies 
shapefiles in a similar way when they are opened in JOSM.

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