You are able to shop at Walmart or Publix at their pleasure. You have no right to be there.
Since it's private property, they can legally ask you to leave. They don't have to give a reason.
If you refuse, they can call the police and have you trespassed.
If you still refuse to leave, or return after having been trespassed, you can be arrested.
Of course Publix or Walmart can ask you to leave. The signs do not do that.
They request you do not open carry. So, when I go into Walmart, I untuck my shirt if it is tucked. Sometimes I have forgotten to do so.
No one has ever asked me to leave. When I see other open carriers, which is not common, I ask them if they have been confronted. As I recall none have said they have been asked to leave. The numbers are small, perhaps half a dozen. Still, it is an indication the policy is to politely ask people not to open carry, not a prohibition on open carry.
In effect, it gives the management a means to ask someone who is acting irresponsibly, to leave. They have always had the ability to do that, whether the sign was there or not.