What specific type of toothpaste can a person along with chlorhexidine gluconate mouthwash?

Any toothpaste. When using a Chlorhexidine mouthwash as an adjunct for oral hygiene, I recommend brushing and flossing first, rinsing with water and then using the mouthwash.
Supersmile. Peridex (chlorhexidine gluconate) mouthwash stains teeth brown. Supersmile toothpaste dissolves the protein pellicle, which prevents Peridex (chlorhexidine gluconate) stain from attaching to your teeth.
Any but rinse well. If you use chlorhexidine, you should rinse your mouth well w/ water after brushing your teeth & before using chlorhexidine. This is b/c some ingredients in toothpaste can inactivate chlorhexidine. Chlorhexidine can also stain teeth brown when used regularly. This staining is likely to need to be removed by a dentist or dental hygienist.