This　paper　attempts　to　study　nonzero-sum　continuous-time　constrained　average　stochastic　games　with　independent　state　processes.　In　these　game　models,　each　player　independently　controls　a　continuous-time　Markov　chain,　but　players　are　coupled　by　the　immediate　cost　functions.　The　transition　rates　and　immediate　cost　functions　are　allowed　to　be　unbounded.　Each　player　wants　to　minimize　certain　expected　average　cost,　but　constraints　are　imposed　on　other　expected　average　costs.　By　introducing　the　average　occupation　measures,　we　establish　the　one-to-one　relationship　of　constrained　Nash　equilibria　and　the　fixed　points　of　certain　multifunction　defined　on　the　product　space　of　average　occupation　measures.　Then,　by　using　the　fixed　point　theorem,　we　show　the　existence　of　constrained　Nash　equilibria.　Finally,　we　show　that　each　stationary　Nash　equilibrium　corresponds　to　a　global　minimizer　of　a　certain　mathematical　program.