Before I try what John suggested, I decided to post the .htaccess files for WordPress and Omeka. Perhaps this will help identify what must be done to resolve any conflicts.
WordPress .htaccess file
ErrorDocument 401 /401.html
AuthUserFile "[path_deleted]"
# BEGIN WordPress
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>
# END WordPress
# BEGIN Custom Security
## Makes it harder to exploit index.html access.
Options All -Indexes
# END Custom Security
Omeka .htaccess file
# Remember to enable display_errors on development environments only.
<IfModule mod_php5.c>
php_value display_errors 0
php_flag register_globals off
</IfModule>
RewriteEngine on
# If know that mod_rewrite is enabled, but you are still getting mod_rewrite errors,
# uncomment the line below and replace "/" with your base directory.
# RewriteBase /
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^admin/ - [C]
RewriteRule .* admin/index.php [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule .* index.php
<FilesMatch "\.(php|ini)$">
Order Allow,Deny
Deny from all
</FilesMatch>
<Files index.php>
Allow from all
</Files>
# Uncomment the lines below in order to enable caching of some files via Apache (after a finished site has gone live)
<IfModule mod_expires.c>
# <FilesMatch "\.(js|ico|gif|jpg|png|css)$">
# ExpiresActive on
# ExpiresDefault "access plus 10 day"
# </FilesMatch>
</IfModule>
