<p>We present the design of two interacting harmonic non-elliptical compressible liquid inclusions embedded in an infinite isotropic elastic matrix subjected to uniform remote in-plane stresses. The original constant mean stress (or the first invariant of the stress tensor) in the matrix remains undisturbed in the presence of the two harmonic liquid inclusions. The two non-elliptical liquid-solid interfaces are described by a four-parameter conformal mapping function that maps the doubly connected domain occupied by the matrix onto an annulus in the image plane. The closed-form expressions for the internal uniform hydrostatic stress fields within the two liquid inclusions are obtained. The hoop stresses are uniformly distributed along the two liquid-solid interfaces on the matrix side.</p>