  It has been shown [1] that many seemingly contradictory experimental findings
concerning the superconducting state in Sr$_2$RuO$_4$ can be accounted for as
resulting from the existence of an assumed tetra-critical point at near ambient
pressure at which $d_{x^2-y^2}$ and $g_{xy(x^2-y^2)}$ superconducting states
are degenerate. We perform both a Landau-Ginzburg and a microscopic mean-field
analysis of the effect of spatially varying strain on such a state. In the
presence of finite $xy$ shear strain, the superconducting state consists of two
possible symmetry-related time-reversal symmetry (TRS) preserving states: $d
\pm g$. However, at domain walls between two such regions, TRS can be broken,
resulting in a $d+ig$ state. More generally, we find that various natural
patterns of spatially varying strain induce a rich variety of superconducting
textures, including half-quantum fluxoids. These results may resolve some of
the apparent inconsistencies between the theoretical proposal and various
experimental observations, including the suggestive evidence of half-quantum
vortices [2].
  [1] Steven A Kivelson, Andrew C Yuan, BJ Ramshaw, and Ronny Thomale, "A
proposal for reconciling diverse experiments on the superconducting state in
Sr$_2$RuO$_4$," npj Quantum Mater 5 (2020).
  [2] J Jang, DG Ferguson, V Vakaryuk, Raffi Budakian, SB Chung, PM Goldbart,
and Y Maeno, "Observation of half-height magnetization steps in Sr$_2$RuO$_4$,"
Science 331, 186-188 (2011).
