  A proto-quantum space is a (general) matricially normed space in the sense of
Effros and Ruan presented in a `matrix-free' language. We show that these
spaces have a special (projective) tensor product possessing the universal
property with respect to completely bounded bilinear operators. We study some
general properties of this tensor product (among them a kind of adjoint
associativity), and compute it for some tensor factors, notably for $L_1$
spaces. In particular, we obtain what could be called the proto-quantum version
of the Grothendieck theorem about classical projective tensor products by $L_1$
spaces. At the end, we compare the new tensor product with the known projective
tensor product of operator spaces, and show that the standard construction of
the latter is not fit for general proto-quantum spaces.
