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Using Lakshmibai-Seshadri paths, we give a combinatorial realization of thecrystal basis of an extremal weight module of integral extremal weight over thequantized universal enveloping algebra associated to the infinite rank affineLie algebra of type $B_{\infty}$, $C_{\infty}$, or $D_{\infty}$. Moreover, viathis realization, we obtain an explicit description (in terms ofLittlewood-Richardson coefficients) of how tensor products of these crystalbases decompose into connected components when their extremal weights are ofnonnegative levels. These results, in types $B_{\infty}$, $C_{\infty}$, and$D_{\infty}$, extend the corresponding results due to Kwon, in types$A_{+\infty}$ and $A_{\infty}$; our results above also include, as a specialcase, the corresponding results (concerning crystal bases) due to Lecouvey, intypes $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$, where the extremal weightsare of level zero.

Tensor product multiplicities for crystal bases of extremal weight  modules over quantum infinite rank affine algebras of types $B_{\infty}$,  $C_{\infty}$, and $D_{\infty}$