Table 1 Relationships between the signs of order parameters.

Condition before measurement	h [Q_A, Q_B]	P _z	X₁ (Γ¹)	X₃=M_z (Γ³)	X₁X₃ (C_A or C_B)	Q (Q_A or Q_B)
+E+B cool	[−, +]	+	+(−)	+	+(−)	Q_A(Q_B)
−B sweep	[+,−]	−	+(−)	−	−(+)	Q_B(Q_A)
+E−B cool	[−, +]	+	−(+)	−	+(−)	Q_A(Q_B)
+B sweep	[+,−]	−	−(+)	+	−(+)	Q_B(Q_A)
−E+B cool	[+, −]	−	+(−)	+	+(−)	Q_A(Q_B)
−B sweep	[−,+]	+	+(−)	−	−(+)	Q_B(Q_A)
−E−B cool	[+, −]	−	−(+)	−	+(−)	Q_A(Q_B)
+B sweep	[−,+]	+	−(+)	+	−(+)	Q_B(Q_A)

The signs are determined by considering that an electric field E and a magnetic field B directly act on P_z and M_z, respectively, and by the U, V and W invariants in equations (1)–(3), . In the table, square brackets show the sign of h in each Q domain. The sign ‘+(−)’ denotes that either ‘+’ or ‘−’ domain is possible. However, within each experimental condition, the signs in and out of parentheses couple with those in and out of parentheses for other order parameters. Here, ‘+E+B cool’ means the field-cooled condition with positive E and B that are large enough to pole the system. ‘B-sweep’ denotes that B was swept to an opposite value once after the cooling procedure of the upper row. One-to-one correspondence between h and P_z is ascribed to the U term (equation (1)) and that between X₁X₃ and Q originates from the V term (equations (2)). In the column of X₁X₃, ‘+’ and ‘−’ refer to C_A and C_B commensurate domains, respectively. The relationships between the respective order parameters satisfy all experimental results (Table 2 and Supplementary Table 2).

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