How can I get Latin Modern/Computer Modern serif double-struck/blackboard-bold characters which copy/paste as unicode when using unicode-math?

Question

I want my characters to copy-paste as Unicode when I copy them from the PDF. I also want them to look like the traditional Latin Modern font. For unrelated reasons, I need to use LuaLaTeX rather than pdflatex. The unicode-math package works fine in most cases, but produces strange sans-serif double-struck characters. How do I fix this?

Example:

\documentclass{article}
\usepackage{amssymb}
\begin{document}
\[\alpha\beta\gamma\pi\sum\prod1+2+3+4+5+6+7+8+9+0/1!\]
\[abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ\]
\[\mathbb{abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ}\]
\end{document}

in lualatex gives

whereas

\documentclass{article}
\usepackage{amssymb}
\usepackage{unicode-math}
\begin{document}
\[\alpha\beta\gamma\pi\sum\prod1+2+3+4+5+6+7+8+9+0/1!\]
\[abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ\]
\[\mathbb{abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ}\]
\end{document}

in lualatex gives

While most of the characters are identical are nearly identical, the capital double-struck characters are clearly not. They don't even have serifs, unlike the rest of the characters. It's cool that I can now get lowercase double-struck characters, but this doesn't make up for the jarring and stylistically very-different uppercase double-struck characters. The answer at https://tex.stackexchange.com/a/360609/2066 gives the correct characters, but then the characters don't copy-paste as double-struck. How do I get the Latin Modern / Computer Modern double-struck characters back without losing the unicode-copy-paste feature of unicode-math?

(I'm also curious about the history here; why are these characters so different? https://tex.stackexchange.com/a/525235/2066 says "Latin Modern Math is a clone of Knuth’s Computer Modern, and unicode-math produces nearly-identical output under the default settings (with a few caveats: the \mathcal alphabet is based on Euler Calligraphic instead of traditional LaTeX).")

Answer 1

The classic TeX double struck font you show in the first code block is not derived from Computer Modern at all it is the AMS symbol font msbm10 (and the first two rows are set in Computer Modern, not Latin Modern)

The second block is using a consistent font for all three rows, Latin Modern Math Opentype.

There are other Computer Modern inspired OpenType Math fonts, notably NewComputerModernMath. You may prefer the double struck math in this font:

\documentclass{article}
\usepackage{amssymb}
\usepackage{unicode-math}
\setmathfont{New Computer Modern Math}
\begin{document}
\[\alpha\beta\gamma\pi\sum\prod1+2+3+4+5+6+7+8+9+0/1!\]
\[abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ\]
\[\mathbb{abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ}\]
\end{document}