embed_resources option breaks slides when output is revealjs and html

I have:

• searched the issue tracker for similar issues
• installed the latest version of Quarto CLI
• formatted my issue following the Bug Reports guide

Bug description

I think this may be related to #7221, I create both my lecture slides and course notes from a single qmd. I use output-file to make sure that the revealjs and html filenames don't clash. When using embed-resources on the html output, formatting on the revealjs slides is broken. Turning off embed_resources in the html output resolves the problem.

This happens on both the current quarto release (1.9.36) and the pre-release I tested (1.10.1). I'm unsure if this is a quarto bug or a revealjs bug.

This is a pretty minor issue, it's very simple to work around. I can just copy the "correct" slides, then regenerate the html with embed-resources, so just providing this report for information.

Steps to reproduce

---
title: Test
format:
  revealjs:
    output-file: test.html
  html:
    theme: darkly 
---

# Test Slide

## Mic Check

Add embed-resources will cause the revealjs output to lose formatting.

---
title: Test
format:
  revealjs:
    output-file: test.html
  html:
    theme: darkly
    embed-resources: true
---

# Test Slide

## Mic Check

Actual behavior

Revealjs slides lose all formatting.

Expected behavior

Revealjs slides should not lose all formatting.

Your environment

No response

Quarto check output

Quarto 1.10.1

[✓] Checking environment information...
Quarto cache location: /home/khonsu/.cache/quarto

[✓] Checking versions of quarto binary dependencies...
Pandoc version 3.8.3: OK
Dart Sass version 1.87.0: OK
Deno version 2.4.5: OK
Typst version 0.14.2: OK

[✓] Checking versions of quarto dependencies......OK

[✓] Checking Quarto installation......OK
Version: 1.10.1
Path: /home/khonsu/applications/quarto/bin

[✓] Checking tools....................OK
TinyTeX: (not installed)
Chromium: (not installed)
Chrome Headless Shell: (not installed)
VeraPDF: (not installed)

[✓] Checking LaTeX....................OK
Using: Installation From Path
Path: /home/khonsu/applications/texlive/2025/bin/x86_64-linux
Version: 2025

[✓] Checking Chrome Headless....................OK
Chrome: (not detected)

[✓] Checking basic markdown render....OK

[✓] Checking R installation...........OK
Version: 4.5.3
Path: /usr/lib64/R
LibPaths:
- /home/khonsu/rlibrary
- /usr/lib64/R/library
knitr: 1.51
rmarkdown: 2.30

[✓] Checking Knitr engine render......OK

[✓] Checking Python 3 installation....OK
Version: 3.13.12
Path: /usr/lib/python-exec/python3.13/python3
Jupyter: (None)

Start by rendering the two qmd examples from the report with Quarto, comparing the revealjs output with and without html embed-resources enabled. Trace the shared output-resource handling and determine why the HTML render affects the revealjs formatting; done means both outputs retain their expected formatting when generated together.