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We study the strong decays of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mo>(</mo><mn>2170</mn><mo>)</mo></mrow></semantics></math></inline-formula>, along with its possible partner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>(</mo><mn>2436</mn><mo>)</mo></mrow></semantics></math></inline-formula>, as two fully strange tetraquark states of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></mrow></semantics></math></inline-formula>. These two states are assumed to contain two strange quarks and two anti-strange quarks, with the flavor symmetry <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mn mathvariant="bold">6</mn><mrow><mi>s</mi><mi>s</mi></mrow></msub><mo>⊗</mo><msub><mover accent="true"><mn mathvariant="bold">6</mn><mo stretchy="false">¯</mo></mover><mrow><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow></msub></mrow></semantics></math></inline-formula>. We consider seven decay channels: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mi>η</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><msup><mi>η</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>980</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><msub><mi>f</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1420</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1415</mn><mo>)</mo></mrow><mi>η</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1415</mn><mo>)</mo></mrow><msup><mi>η</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1415</mn><mo>)</mo></mrow><msub><mi>f</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1420</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Some of these channels are kinematically possible, and we calculate their relative branching ratios through the Fierz rearrangement. Future experimental measurements on these ratios could be useful in determining the nature of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mo>(</mo><mn>2170</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>(</mo><mn>2436</mn><mo>)</mo></mrow></semantics></math></inline-formula>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mo>(</mo><mn>2170</mn><mo>)</mo></mrow></semantics></math></inline-formula> has been observed in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>980</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mi>η</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><msup><mi>η</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula> channels, and we propose to further examine it in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1415</mn><mo>)</mo></mrow><mi>η</mi></mrow></semantics></math></inline-formula> channel. Evidences of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>(</mo><mn>2436</mn><mo>)</mo></mrow></semantics></math></inline-formula> have been observed in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>980</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> channel, and we propose to verify whether this structure exists or not in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mi>η</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><msup><mi>η</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1415</mn><mo>)</mo></mrow><mi>η</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1415</mn><mo>)</mo></mrow><msup><mi>η</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula> channels.