if so, we need a tag synonym:
No, they're not the same.
Spider mites are members of the Acari (mite) family Tetranychidae, which includes about 1,200 species. They generally live on the under sides of leaves of plants, where they may spin protective silk webs, and they can cause damage by puncturing the plant cells to feed. Spider mites are known to feed on several hundred species of plant.
Dermanyssus gallinae, also known as red mite, is an ectoparasite of poultry and birds in general. The mites are blood feeders and attack resting birds at night. After feeding, they hide in cracks and crevices away from daylight, where they mate and lay eggs. Under favourable conditions the life cycle of the mite can be completed within seven days, so populations can grow rapidly, causing anaemia in badly affected flocks of poultry.
There are no synonyms in the system for either spider-mites or red-mites. So it is likely that it was tagged incorrectly by mistake (possibly because OP himself refers to it as red mites in his last sentence). While it's true that spider mites look red in colour, they're not the same as red mites.