Let K/Q be a real quadratic field. Given an automorphic representation π for GL2/K, let As±(π) denote the plus/minus Asai transfer of π to an automorphic representation for GL4/Q. In this paper, we construct a rigid analytic map from a subvariety of the universal eigenvariety of GL2/K to an eigenvariety of GL4/Q, which at nice classical points interpolate this Asai transfer. © 2019 Ramanujan Mathematical Society. All rights reserved.