Technically yes its illegal. But Police are unlikely to enforce it without some form of criminal intent or property damage or circumvention of locks or ignoring of clearly posted "no trespassing" signs.

The grey area is whether going into someone's back yard constitutes criminal trespassing. For instance you can knock on someone's front door and although you are 100% trespassing on their property the moment you step foot on their front yard, you cant really be charged with criminal trespassing unless they ask you to leave and you refuse to do so, or if the owner tells them they aren't welcomed on the property and they return knowing they have no consent to do so.

Legally the front yard is no different than the back yard, its all the same piece of private property under the law. Stepping into their front lawn is just as much trespassing as stepping into their back yard. The only distinction is perhaps a gate which should imply to people that they have no consent to go past this point. My recommendation to the OP was to remove any ambiguity as to whether the neighbor could claim ignorance as to whether or not they had implied consent to go through that gate, to make it easier for police to charge the neighbor with a crime. Best ways are clear "no trespassing signs" and locks or barriers.

While any trespassing on private property is illegal. You also need to make things easy enough for the police to enforce the law and courts to prosecute, otherwise the police will likely just ignore it. Because they're looking for laws that will actually be prosecuted. An old nosy neighbor lady wandering into someone's backyard probably wont make it to court, and probably wont lead anywhere. But the same actions for instance by a complete stranger with no explanation for the trespassing might though, because its possible to even push for attempted burglary depending on the circumstances. A DA, judge, and jury will see these two individuals very differently. Ones just a nosey old lady and the other is a potential burglar, despite their actions being the same. But how they appear and their intents greatly change how they perceived. And police tend to avoid enforcing crimes that they know will just end up wasting the court system's time. But if you lock your gates and put up signs and cameras, you give the police crimes that they can enforce, crimes that could be prosecuted and not be thrown out, evidence that would stand up in court, things that would sway a jury, etc... Because now you'll have evidence of the nosey neighbor taking extreme willfull actions to force entry onto private property with complete disregard for signs forbidding access.