  In this paper, we first define the multivariate tempered space-fractional
Poisson process (MTSFPP) by time-changing the multivariate Poisson process with
an independent tempered {\alpha}-stable subordinator. Its distributional
properties, the mixture tempered time and space variants and their PDEs
connections are studied. Then we define the multivariate tempered
space-fractional negative binomial process (MTSFNBP) and explore its key
features. The L\'evy measure density for the MTSFNBP is also derived. We
present a bivariate risk model with a common shock driven by the tempered
space-fractional negative binomial process. We demonstrate that the total claim
amount process is stochastically equivalent to a univariate generalized
Cramer-Lundberg risk model. In addition, some important ruin measures such as
ruin probability, joint distribution of time to ruin and deficit at ruin along
with governing integro-differential equations are obtained. Finally we show
that the underlying risk process exhibits the long range dependence property.
